Stochastic Systems Articles (Project Euclid)
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The latest articles from Stochastic Systems on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2014 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Tue, 11 Feb 2014 10:10 ESTTue, 11 Feb 2014 10:10 ESThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Tuning approximate dynamic programming policies for ambulance redeployment via direct search
http://projecteuclid.org/euclid.ssy/1392131419
<strong>Matthew S. Maxwell</strong>, <strong>Shane G. Henderson</strong>, <strong>Huseyin Topaloglu</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 3, Number 2, 322--361.</p><p><strong>Abstract:</strong><br/>
In this paper we consider approximate dynamic programming methods for ambulance redeployment. We first demonstrate through simple examples how typical value function fitting techniques, such as approximate policy iteration and linear programming, may not be able to locate a high-quality policy even when the value function approximation architecture is rich enough to provide the optimal policy. To make up for this potential shortcoming, we show how to use direct search methods to tune the parameters in a value function approximation architecture so as to obtain high-quality policies. Direct search is computationally intensive. We therefore use a post-decision state dynamic programming formulation of ambulance redeployment that, together with direct search, requires far less computation with no noticeable performance loss. We provide further theoretical support for the post-decision state formulation of the ambulance-deployment problem by showing that this formulation can be obtained through a limiting argument on the original dynamic programming formulation.
</p>projecteuclid.org/euclid.ssy/1392131419_20140211101024Tue, 11 Feb 2014 10:10 ESTStability of a stochastic model for demand-responsehttp://projecteuclid.org/euclid.ssy/1393251980<strong>Jean-Yves Le Boudec</strong>, <strong>Dan-Cristian Tomozei</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 3, Number 1, 11--37.</p><p><strong>Abstract:</strong><br/>
We study the stability of a Markovian model of electricity production and consumption that incorporates production volatility due to renewables and uncertainty about actual demand versus planned production. We assume that the energy producer targets a fixed energy reserve, subject to ramp-up and ramp-down constraints, and that appliances are subject to demand-response signals and adjust their consumption to the available production by delaying their demand. When a constant fraction of the delayed demand vanishes over time, we show that the general state Markov chain characterizing the system is positive Harris and ergodic (i.e., delayed demand is bounded with high probability). However, when delayed demand increases by a constant fraction over time, we show that the Markov chain is non-positive (i.e., there exists a non-zero probability that delayed demand becomes unbounded). We exhibit Lyapunov functions to prove our claims. In addition, we provide examples of heating appliances that, when delayed, have energy requirements corresponding to the two considered cases.
</p>projecteuclid.org/euclid.ssy/1393251980_20140224092623Mon, 24 Feb 2014 09:26 ESTOn the convergence of simulation-based iterative methods for solving singular linear systemshttp://projecteuclid.org/euclid.ssy/1393251981<strong>Mengdi Wang</strong>, <strong>Dimitri P. Bertsekas</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 3, Number 1, 38--95.</p><p><strong>Abstract:</strong><br/>
We consider the simulation-based solution of linear systems of equations, $Ax=b$, of various types frequently arising in large-scale applications, where $A$ is singular. We show that the convergence properties of iterative solution methods are frequently lost when they are implemented with simulation (e.g., using sample average approximation), as is often done in important classes of large-scale problems. We focus on special cases of algorithms for singular systems, including some arising in least squares problems and approximate dynamic programming, where convergence of the residual sequence $\{Ax_{k}-b\}$ may be obtained, while the sequence of iterates $\{x_{k}\}$ may diverge. For some of these special cases, under additional assumptions, we show that the iterate sequence is guaranteed to converge. For situations where the iterates diverge but the residuals converge to zero, we propose schemes for extracting from the divergent sequence another sequence that converges to a solution of $Ax=b$.
</p>projecteuclid.org/euclid.ssy/1393251981_20140224092623Mon, 24 Feb 2014 09:26 ESTMany-server queues with customer abandonment: Numerical analysis of their diffusion modelhttp://projecteuclid.org/euclid.ssy/1393251982<strong>J. G. Dai</strong>, <strong>Shuangchi He</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 3, Number 1, 96--146.</p><p><strong>Abstract:</strong><br/>
We use a multidimensional diffusion process to approximate the dynamics of a queue served by many parallel servers. Waiting customers in this queue may abandon the system without service. To analyze the diffusion model, we develop a numerical algorithm for computing its stationary distribution. A crucial part of the algorithm is choosing an appropriate reference density. Using a conjecture on the tail behavior of the limit queue length process, we propose a systematic approach to constructing a reference density. With the proposed reference density, the algorithm is shown to converge quickly in numerical experiments. These experiments demonstrate that the diffusion model is a satisfactory approximation for many-server queues, sometimes for queues with as few as twenty servers.
</p>projecteuclid.org/euclid.ssy/1393251982_20140224092623Mon, 24 Feb 2014 09:26 ESTDirected random graphs with given degree distributionshttp://projecteuclid.org/euclid.ssy/1393251983<strong>Ningyuan Chen</strong>, <strong>Mariana Olvera-Cravioto</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 3, Number 1, 147--186.</p><p><strong>Abstract:</strong><br/>
Given two distributions $F$ and $G$ on the nonnegative integers we propose an algorithm to construct in- and out-degree sequences from samples of i.i.d. observations from $F$ and $G$, respectively, that with high probability will be graphical, that is, from which a simple directed graph can be drawn. We then analyze a directed version of the configuration model and show that, provided that $F$ and $G$ have finite variance, the probability of obtaining a simple graph is bounded away from zero as the number of nodes grows. We show that conditional on the resulting graph being simple, the in- and out-degree distributions are (approximately) $F$ and $G$ for large size graphs. Moreover, when the degree distributions have only finite mean we show that the elimination of self-loops and multiple edges does not significantly change the degree distributions in the resulting simple graph.
</p>projecteuclid.org/euclid.ssy/1393251983_20140224092623Mon, 24 Feb 2014 09:26 ESTOptimal paths in large deviations of symmetric reflected Brownian motion in the octanthttp://projecteuclid.org/euclid.ssy/1393251984<strong>Ziyu Liang</strong>, <strong>John J. Hasenbein</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 3, Number 1, 187--229.</p><p><strong>Abstract:</strong><br/>
We study the variational problem that arises from consideration of large deviations for semimartingale reflected Brownian motion (SRBM) in $\mathbb{R}^{3}_{+}$. Due to the difficulty of the general problem, we consider the case in which the SRBM has rotationally symmetric parameters. In this case, we are able to obtain conditions under which the optimal solutions to the variational problem are paths that are gradual (moving through faces of strictly increasing dimension) or that spiral around the boundary of the octant. Furthermore, these results allow us to provide an example for which it can be verified that a spiral path is optimal. For rotationally symmetric SRBM’s, our results facilitate the simplification of computational methods for determining optimal solutions to variational problems and give insight into large deviations behavior of these processes.
</p>projecteuclid.org/euclid.ssy/1393251984_20140224092623Mon, 24 Feb 2014 09:26 ESTA linear response bandit problemhttp://projecteuclid.org/euclid.ssy/1393251985<strong>Alexander Goldenshluger</strong>, <strong>Assaf Zeevi</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 3, Number 1, 230--261.</p><p><strong>Abstract:</strong><br/>
We consider a two–armed bandit problem which involves sequential sampling from two non-homogeneous populations. The response in each is determined by a random covariate vector and a vector of parameters whose values are not known a priori. The goal is to maximize cumulative expected reward. We study this problem in a minimax setting, and develop rate-optimal polices that combine myopic action based on least squares estimates with a suitable “forced sampling” strategy. It is shown that the regret grows logarithmically in the time horizon $n$ and no policy can achieve a slower growth rate over all feasible problem instances. In this setting of linear response bandits, the identity of the sub-optimal action changes with the values of the covariate vector, and the optimal policy is subject to sampling from the inferior population at a rate that grows like $\sqrt{n}$.
</p>projecteuclid.org/euclid.ssy/1393251985_20140224092623Mon, 24 Feb 2014 09:26 ESTFluid limits for overloaded multiclass FIFO single-server queues with general abandonmenthttp://projecteuclid.org/euclid.ssy/1393251986<strong>Otis B. Jennings</strong>, <strong>Amber L. Puha</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 3, Number 1, 262--321.</p><p><strong>Abstract:</strong><br/>
We consider an overloaded multiclass nonidling first-in-first-out single-server queue with abandonment. The interarrival times, service times, and deadline times are sequences of independent and identically, but generally distributed random variables. In prior work, Jennings and Reed studied the workload process associated with this queue. Under mild conditions, they establish both a functional law of large numbers and a functional central limit theorem for this process. We build on that work here. For this, we consider a more detailed description of the system state given by $K$ finite, nonnegative Borel measures on the nonnegative quadrant, one for each job class. For each time and job class, the associated measure has a unit atom associated with each job of that class in the system at the coordinates determined by what are referred to as the residual virtual sojourn time and residual patience time of that job. Under mild conditions, we prove a functional law of large numbers for this measure-valued state descriptor. This yields approximations for related processes such as the queue lengths and abandoning queue lengths. An interesting characteristic of these approximations is that they depend on the deadline distributions in their entirety.
</p>projecteuclid.org/euclid.ssy/1393251986_20140224092623Mon, 24 Feb 2014 09:26 ESTDistributed user profiling via spectral methodshttp://projecteuclid.org/euclid.ssy/1411044991<strong>Dan-Cristian Tomozei</strong>, <strong>Laurent Massoulié</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 1, 1--43.</p><p><strong>Abstract:</strong><br/>
User profiling is a useful primitive for constructing personalised services, such as content recommendation. In the present paper we investigate the feasibility of user profiling in a distributed setting, with no central authority and only local information exchanges between users. We compute a profile vector for each user (i.e., a low-dimensional vector that characterises her taste) via spectral transformation of observed user-produced ratings for items. Our two main contributions follow:
(i) We consider a low-rank probabilistic model of user taste. More specifically, we consider that users and items are partitioned in a constant number of classes, such that users and items within the same class are statistically identical. We prove that without prior knowledge of the compositions of the classes, based solely on few random observed ratings (namely $O(N\log N)$ such ratings for $N$ users), we can predict user preference with high probability for unrated items by running a local vote among users with similar profile vectors. In addition, we provide empirical evaluations characterising the way in which spectral profiling performance depends on the dimension of the profile space. Such evaluations are performed on a data set of real user ratings provided by Netflix.
(ii) We develop distributed algorithms which provably achieve an embedding of users into a low-dimensional space, based on spectral transformation. These involve simple message passing among users, and provably converge to the desired embedding. Our method essentially relies on a novel combination of gossiping and the algorithm proposed by Oja and Karhunen.
</p>projecteuclid.org/euclid.ssy/1411044991_20140918085634Thu, 18 Sep 2014 08:56 EDTDeterministic and stochastic primal-dual subgradient algorithms for uniformly convex minimizationhttp://projecteuclid.org/euclid.ssy/1411044992<strong>Anatoli Juditsky</strong>, <strong>Yuri Nesterov</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 1, 44--80.</p><p><strong>Abstract:</strong><br/>
We discuss non-Euclidean deterministic and stochastic algorithms for optimization problems with strongly and uniformly convex objectives. We provide accuracy bounds for the performance of these algorithms and design methods which are adaptive with respect to the parameters of strong or uniform convexity of the objective: in the case when the total number of iterations $N$ is fixed, their accuracy coincides, up to a logarithmic in $N$ factor with the accuracy of optimal algorithms.
</p>projecteuclid.org/euclid.ssy/1411044992_20140918085634Thu, 18 Sep 2014 08:56 EDTQueue-based random-access algorithms: Fluid limits and stability issueshttp://projecteuclid.org/euclid.ssy/1411044993<strong>Javad Ghaderi</strong>, <strong>Sem Borst</strong>, <strong>Phil Whiting</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 1, 81--156.</p><p><strong>Abstract:</strong><br/>
We use fluid limits to explore the (in)stability properties of wireless networks with queue-based random-access algorithms. Queue-based random-access schemes are simple and inherently distributed in nature, yet provide the capability to match the optimal throughput performance of centralized scheduling mechanisms in a wide range of scenarios. Unfortunately, the type of activation rules for which throughput optimality has been established, may result in excessive queue lengths and delays. The use of more aggressive/persistent access schemes can improve the delay performance, but does not offer any universal maximum-stability guarantees.
In order to gain qualitative insight and investigate the (in)stability properties of more aggressive/persistent activation rules, we examine fluid limits where the dynamics are scaled in space and time. In some situations, the fluid limits have smooth deterministic features and maximum stability is maintained, while in other scenarios they exhibit random oscillatory characteristics, giving rise to major technical challenges. In the latter regime, more aggressive access schemes continue to provide maximum stability in some networks, but may cause instability in others. In order to prove that, we focus on a particular network example and conduct a detailed analysis of the fluid limit process for the associated Markov chain. Specifically, we develop a novel approach based on stopping time sequences to deal with the switching probabilities governing the sample paths of the fluid limit process. Simulation experiments are conducted to illustrate and validate the analytical results.
</p>projecteuclid.org/euclid.ssy/1411044993_20140918085634Thu, 18 Sep 2014 08:56 EDTPower identities for Lévy risk models under taxation and capital injectionshttp://projecteuclid.org/euclid.ssy/1411044994<strong>Hansjörg Albrecher</strong>, <strong>Jevgenijs Ivanovs</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 1, 157--172.</p><p><strong>Abstract:</strong><br/>
In this paper we study a spectrally negative Lévy process which is refracted at its running maximum and at the same time reflected from below at a certain level. Such a process can for instance be used to model an insurance surplus process subject to tax payments according to a loss-carry-forward scheme together with the flow of minimal capital injections required to keep the surplus process non-negative. We characterize the first passage time over an arbitrary level and the cumulative amount of injected capital up to this time by their joint Laplace transform, and show that it satisfies a simple power relation to the case without refraction, generalizing results by Albrecher and Hipp (2007) and Albrecher, Renaud and Zhou (2008). It turns out that this identity can also be extended to a certain type of refraction from below. The net present value of tax collected before the cumulative injected capital exceeds a certain amount is determined, and a numerical illustration is provided.
</p>projecteuclid.org/euclid.ssy/1411044994_20140918085634Thu, 18 Sep 2014 08:56 EDTLarge deviations of the interference in the Ginibre network modelhttp://projecteuclid.org/euclid.ssy/1411044995<strong>Giovanni Luca Torrisi</strong>, <strong>Emilio Leonardi</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 1, 173--205.</p><p><strong>Abstract:</strong><br/>
Under different assumptions on the distribution of the fading random variables, we derive large deviation estimates for the tail of the interference in a wireless network model whose nodes are placed, over a bounded region of the plane, according to the $\beta$-Ginibre process, $0<\beta\leq1$. The family of $\beta$-Ginibre processes is formed by determinantal point processes, with different degree of repulsiveness. As $\beta\to0$, $\beta$-Ginibre processes converge in law to a homogeneous Poisson process. In this sense the Poisson network model may be considered as the limiting uncorrelated case of the $\beta$-Ginibre network model. Our results indicate the existence of two different regimes.
When the fading random variables are bounded or Weibull superexponential, large values of the interference are typically originated by the sum of several equivalent interfering contributions due to nodes in the vicinity of the receiver. In this case, the tail of the interference has, on the log-scale, the same asymptotic behavior for any value of $0<\beta\le1$, but it differs from the asymptotic behavior of the tail of the interference in the Poisson network model (again on a log-scale) [14].
When the fading random variables are exponential or subexponential, instead, large values of the interference are typically originated by a single dominating interferer node and, on the log-scale, the asymptotic behavior of the tail of the interference is insensitive to the distribution of the nodes, as long as the number of nodes is guaranteed to be light-tailed.
</p>projecteuclid.org/euclid.ssy/1411044995_20140918085634Thu, 18 Sep 2014 08:56 EDTTwo-parameter sample path large deviations for infinite-server queueshttp://projecteuclid.org/euclid.ssy/1411044996<strong>Jose Blanchet</strong>, <strong>Xinyun Chen</strong>, <strong>Henry Lam</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 1, 206--249.</p><p><strong>Abstract:</strong><br/>
Let $Q_{\lambda}(t,y)$ be the number of people present at time $t$ with at least $y$ units of remaining service time in an infinite server system with arrival rate equal to $\lambda>0$. In the presence of a non-lattice renewal arrival process and assuming that the service times have a continuous distribution, we obtain a large deviations principle for $Q_{\lambda}(\cdot)/\lambda$ under the topology of uniform convergence on $[0,T]\times[0,\infty)$. We illustrate our results by obtaining the most likely paths, represented as surfaces, to overflow in the setting of loss queues, and also to ruin in life insurance portfolios.
</p>projecteuclid.org/euclid.ssy/1411044996_20140918085634Thu, 18 Sep 2014 08:56 EDTA skill based parallel service system under FCFS-ALIS — steady state, overloads, and abandonmentshttp://projecteuclid.org/euclid.ssy/1411044997<strong>Ivo Adan</strong>, <strong>Gideon Weiss</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 1, 250--299.</p><p><strong>Abstract:</strong><br/>
We consider a queueing system with $J$ parallel servers $\mathcal{S} =\{m_{1},\ldots,m_{J}\}$, and with customer types $\mathcal{C} =\{a,b,\ldots\}$. A bipartite graph $G$ describes which pairs of server-customer types are compatible. We consider FCFS-ALIS policy: A server always picks the first, longest waiting compatible customer, and a customer is always assigned to the longest idle compatible server. We assume Poisson arrivals and server dependent exponential service times. We derive an explicit product-form expression for the stationary distribution of this system when service capacity is sufficient. We also calculate fluid limits of the system under overload, to show that local steady state exists. We distinguish the case of complete resource pooling when all the customers are served at the same rate by the pooled servers, and the case when the system has a unique decomposition into subsets of customer types, each of which is served at its own rate by a pooled subset of the servers. Finally, we discuss possible behavior of the system with generally distributed abandonments, under many server scaling. This paper complements and extends previous results of Kaplan, Caldentey and Weiss [18], and of Whitt and Talreja [34], as well as previous results of the authors [4, 35] on this topic.
</p>projecteuclid.org/euclid.ssy/1411044997_20140918085634Thu, 18 Sep 2014 08:56 EDTLarge deviation asymptotics for busy periodshttp://projecteuclid.org/euclid.ssy/1411044998<strong>Ken R. Duffy</strong>, <strong>Sean P. Meyn</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 1, 300--319.</p><p><strong>Abstract:</strong><br/>
The busy period for a queue is cast as the area swept under the random walk until it first returns to zero. Encompassing non-i.i.d. increments, the large-deviations asymptotics of the busy period $B$ is addressed, under the assumption that the increments satisfy standard conditions, including a negative drift. The main conclusions provide insight on the probability of a large busy period, and the manner in which this occurs. The scaled probability of a large busy period has the asymptote, for any $b>0$, \[\begin{aligned}&\lim_{n\to\infty}\frac{1}{\sqrt{n}}\log P(B\geq bn)=-K\sqrt{b},\\[3pt]\hbox{where}\quad K=2&\sqrt{-\int_{0}^{\lambda^{*}}\Lambda(\theta)\,d\theta},\quad \hbox{with }\lambda^{*}=\sup\{\theta:\Lambda(\theta)\leq0\},\end{aligned}\] and with $\Lambda$ denoting the scaled cumulant generating function of the increments process. The most likely path to a large swept area is found to be a simple rescaling of the path on $[0,1]$ given by \[\psi^{*}(t)=-\Lambda(\lambda^{*}(1-t))/\lambda^{*}.\] In contrast to the piecewise linear most likely path leading the random walk to hit a high level, this is strictly concave in general. While these two most likely paths have distinctly different forms, their derivatives coincide at the start of their trajectories, and at their first return to zero.
These results partially answer an open problem of Kulick and Palmowski [18] regarding the tail of the work done during a busy period at a single server queue. The paper concludes with applications of these results to the estimation of the busy period statistics $(\lambda^{*},K)$ based on observations of the increments, offering the possibility of estimating the likelihood of a large busy period in advance of observing one.
</p>projecteuclid.org/euclid.ssy/1411044998_20140918085634Thu, 18 Sep 2014 08:56 EDTState-independent importance sampling for random walks with regularly varying incrementshttp://projecteuclid.org/euclid.ssy/1427462419<strong>Karthyek R. A. Murthy</strong>, <strong>Sandeep Juneja</strong>, <strong>Jose Blanchet</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 2, 321--374.</p><p><strong>Abstract:</strong><br/>
We develop importance sampling based efficient simulation techniques for three commonly encountered rare event probabilities associated with random walks having i.i.d. regularly varying increments; namely, 1) the large deviation probabilities, 2) the level crossing probabilities, and 3) the level crossing probabilities within a regenerative cycle. Exponential twisting based state-independent methods, which are effective in efficiently estimating these probabilities for light-tailed increments are not applicable when the increments are heavy-tailed. To address the latter case, more complex and elegant state-dependent efficient simulation algorithms have been developed in the literature over the last few years. We propose that by suitably decomposing these rare event probabilities into a dominant and further residual components, simpler state-independent importance sampling algorithms can be devised for each component resulting in composite unbiased estimators with desirable efficiency properties. When the increments have infinite variance, there is an added complexity in estimating the level crossing probabilities as even the well known zero-variance measures have an infinite expected termination time. We adapt our algorithms so that this expectation is finite while the estimators remain strongly efficient. Numerically, the proposed estimators perform at least as well, and sometimes substantially better than the existing state-dependent estimators in the literature.
</p>projecteuclid.org/euclid.ssy/1427462419_20150327092024Fri, 27 Mar 2015 09:20 EDTMean square convergence rates for maximum quasi-likelihood estimatorshttp://projecteuclid.org/euclid.ssy/1427462420<strong>Arnoud V. den Boer</strong>, <strong>Bert Zwart</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 2, 375--403.</p><p><strong>Abstract:</strong><br/>
In this note we study the behavior of maximum quasilikelihood estimators (MQLEs) for a class of statistical models, in which only knowledge about the first two moments of the response variable is assumed. This class includes, but is not restricted to, generalized linear models with general link function. Our main results are related to guarantees on existence, strong consistency and mean square convergence rates of MQLEs. The rates are obtained from first principles and are stronger than known a.s. rates. Our results find important application in sequential decision problems with parametric uncertainty arising in dynamic pricing.
</p>projecteuclid.org/euclid.ssy/1427462420_20150327092024Fri, 27 Mar 2015 09:20 EDTA two-dimensional, two-sided Euler inversion algorithm with computable error bounds and its financial applicationshttp://projecteuclid.org/euclid.ssy/1427462421<strong>Ning Cai</strong>, <strong>Chao Shi</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 2, 404--448.</p><p><strong>Abstract:</strong><br/>
In this paper we propose an inversion algorithm with computable error bounds for two-dimensional, two-sided Laplace transforms. The algorithm consists of two discretization parameters and two truncation parameters. Based on the computable error bounds, we can select these parameters appropriately to achieve any desired accuracy. Hence this algorithm is particularly useful to provide benchmarks. In many cases, the error bounds decay quickly (e.g., exponentially), making the algorithm very efficient. We apply this algorithm to price exotic options such as spread options and barrier options under various asset pricing models as well as to evaluate the joint cumulative distribution functions of related state variables. The numerical examples indicate that the inversion algorithm is accurate, fast and easy to implement.
</p>projecteuclid.org/euclid.ssy/1427462421_20150327092024Fri, 27 Mar 2015 09:20 EDTBandwidth sharing networks with multiscale traffichttp://projecteuclid.org/euclid.ssy/1427462422<strong>Mathieu Feuillet</strong>, <strong>Matthieu Jonckheere</strong>, <strong>Balakrishna Prabhu</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 2, 449--478.</p><p><strong>Abstract:</strong><br/>
In multi-class communication networks, traffic surges due to one class of users can significantly degrade the performance for other classes. During these transient periods, it is thus of crucial importance to implement priority mechanisms that conserve the quality of service experienced by the affected classes, while ensuring that the temporarily unstable class is not entirely neglected. In this paper, we examine the complex interaction occurring between several classes of traffic when classes obtain bandwidth proportionally to their incoming traffic. We characterize the evolution of the performance measures of the network from the moment the initial surge takes place until the system reaches its equilibrium. Using a time-space-transition-scaling, we show that the trajectories of the temporarily unstable class can be described by a differential equation, while those of the stable classes retain their stochastic nature. In particular, we show that the temporarily unstable class evolves at a time-scale which is much slower than that of the stable classes. Although the time-scales decouple, the dynamics of the temporarily unstable and the stable classes continue to influence one another. We further proceed to characterize the obtained differential equations for several simple network examples. In particular, the macroscopic asymptotic behavior of the unstable class allows us to gain important qualitative insights on how the bandwidth allocation affects performance. We illustrate these results on several toy examples and we finally build a penalization rule using these results for a network integrating streaming and surging elastic traffic.
</p>projecteuclid.org/euclid.ssy/1427462422_20150327092024Fri, 27 Mar 2015 09:20 EDTOn the dynamic control of matching queueshttp://projecteuclid.org/euclid.ssy/1427462423<strong>Itai Gurvich</strong>, <strong>Amy Ward</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 2, 479--523.</p><p><strong>Abstract:</strong><br/>
We consider the optimal control of matching queues with random arrivals. In this model, items arrive to dedicated queues, and wait to be matched with items from other (possibly multiple) queues. A match type corresponds to the set of item classes required for a match. Once a decision has been made to perform a match, the matching itself is instantaneous and the matched items depart from the system. We consider the problem of minimizing finite-horizon cumulative holding costs. The controller must decide which matchings to execute given multiple options. In principle, the controller may choose to wait until some “inventory” of items builds up to facilitate more profitable matches in the future.
We introduce a multi-dimensional imbalance process, that at each time $t$, is given by a linear function of the cumulative arrivals to each of the item classes. A non-zero value of the imbalance at time $t$ means that no control could have matched all the items that arrived by time $t$. A lower bound based on the imbalance process can be specified, at each time point, by a solution to an optimization problem with linear constraints. While not achievable in general, this lower bound can be asymptotically approached under a dedicated item condition (an analogue of the local traffic condition in bandwidth sharing networks). We devise a myopic discrete-review matching control that asymptotically–as the arrival rates become large–achieves the imbalance-based lower bound.
</p>projecteuclid.org/euclid.ssy/1427462423_20150327092024Fri, 27 Mar 2015 09:20 EDTResource sharing networks: Overview and an open problemhttp://projecteuclid.org/euclid.ssy/1427462424<strong>J. Michael Harrison</strong>, <strong>Chinmoy Mandayam</strong>, <strong>Devavrat Shah</strong>, <strong>Yang Yang</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 2, 524--555.</p><p><strong>Abstract:</strong><br/>
This paper provides an overview of the resource sharing networks introduced by Massoulié and Roberts [20] to model the dynamic behavior of Internet flows. Striving to separate the model class from the applications that motivated its development, we assume no prior knowledge of communication networks. The paper also presents an open problem, along with simulation results, a formal analysis, and a selective literature review that provide context and motivation. The open problem is to devise a policy for dynamic resource allocation that achieves what we call hierarchical greedy ideal (HGI) performance in the heavy traffic limit. The existence of such a policy is suggested by formal analysis of an approximating Brownian control problem, assuming that there is “local traffic” on each processing resource.
</p>projecteuclid.org/euclid.ssy/1427462424_20150327092024Fri, 27 Mar 2015 09:20 EDTAn asymptotic optimality result for the multiclass queue with finite buffers in heavy traffichttp://projecteuclid.org/euclid.ssy/1427462425<strong>Rami Atar</strong>, <strong>Mark Shifrin</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 4, Number 2, 556--603.</p><p><strong>Abstract:</strong><br/>
For a multiclass G/G/1 queue with finite buffers, admission and scheduling control, and holding and rejection costs, we construct a policy that is asymptotically optimal in the heavy traffic limit. The policy is specified in terms of a single parameter which constitutes the free boundary point from the Harrison-Taksar free boundary problem, but otherwise depends explicitly on the problem data. The $c\mu$ priority rule is also used by the policy, but in a way that is novel, and, in particular, different than that used in problems with infinite buffers. We also address an analogous problem where buffer constraints are replaced by throughput time constraints.
</p>projecteuclid.org/euclid.ssy/1427462425_20150327092024Fri, 27 Mar 2015 09:20 EDTDiffusion models for double-ended queues with renewal arrival processeshttp://projecteuclid.org/euclid.ssy/1450879280<strong>Xin Liu</strong>, <strong>Qi Gong</strong>, <strong>Vidyadhar G. Kulkarni</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 5, Number 1, 1--61.</p><p><strong>Abstract:</strong><br/>
We study a double-ended queue where buyers and sellers arrive to conduct trades. When there is a pair of buyer and seller in the system, they immediately transact a trade and leave. Thus there cannot be a non-zero number of buyers and sellers simultaneously in the system. We assume that sellers and buyers arrive at the system according to independent renewal processes, and they would leave the system after independent exponential patience times. We establish fluid and diffusion approximations for the queue length process under a suitable asymptotic regime. The fluid limit is the solution of an ordinary differential equation, and the diffusion limit is a time-inhomogeneous asymmetric Ornstein-Uhlenbeck process (O-U process). A heavy traffic analysis is also developed, and the diffusion limit in the stronger heavy traffic regime is a time-homogeneous asymmetric O-U process. The limiting distributions of both diffusion limits are obtained. We also show the interchange of the heavy traffic and steady state limits.
</p>projecteuclid.org/euclid.ssy/1450879280_20151223090124Wed, 23 Dec 2015 09:01 ESTThe morphing of fluid queues into Markov-modulated Brownian motionhttp://projecteuclid.org/euclid.ssy/1450879281<strong>Guy Latouche</strong>, <strong>Giang T. Nguyen</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 5, Number 1, 62--86.</p><p><strong>Abstract:</strong><br/>
Ramaswami showed recently that standard Brownian motion arises as the limit of a family of Markov-modulated linear fluid processes. We pursue this analysis with a fluid approximation for Markov-modulated Brownian motion. We follow a Markov-renewal approach and we prove that the stationary distribution of a Markov-modulated Brownian motion reflected at zero is the limit from the well-analyzed stationary distribution of approximating linear fluid processes. Thus, we provide a new approach for obtaining the stationary distribution of a reflected MMBM without time-reversal or solving partial differential equations. Our results open the way to the analysis of more complex Markov-modulated processes.
Key matrices in the limiting stationary distribution are shown to be solutions of a matrix-quadratic equation, and we describe how this equation can be efficiently solved.
</p>projecteuclid.org/euclid.ssy/1450879281_20151223090124Wed, 23 Dec 2015 09:01 ESTModerate deviations for recursive stochastic algorithmshttp://projecteuclid.org/euclid.ssy/1450879282<strong>Paul Dupuis</strong>, <strong>Dane Johnson</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 5, Number 1, 87--119.</p><p><strong>Abstract:</strong><br/>
We prove a moderate deviation principle for the continuous time interpolation of discrete time recursive stochastic processes. The methods of proof are somewhat different from the corresponding large deviation result, and in particular the proof of the upper bound is more complicated.
</p>projecteuclid.org/euclid.ssy/1450879282_20151223090124Wed, 23 Dec 2015 09:01 ESTControl of parallel non-observable queues: Asymptotic equivalence and optimality of periodic policieshttp://projecteuclid.org/euclid.ssy/1450879283<strong>Jonatha Anselmi</strong>, <strong>Bruno Gaujal</strong>, <strong>Tommaso Nesti</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 5, Number 1, 120--145.</p><p><strong>Abstract:</strong><br/>
We consider a queueing system composed of a dispatcher that routes jobs to a set of non-observable queues working in parallel. In this setting, the fundamental problem is which policy should the dispatcher implement to minimize the stationary mean waiting time of the incoming jobs. We present a structural property that holds in the classic scaling of the system where the network demand (arrival rate of jobs) grows proportionally with the number of queues. Assuming that each queue of type $r$ is replicated $k$ times, we consider a set of policies that are periodic with period $k\sum_{r}p_{r}$ and such that exactly $p_{r}$ jobs are sent in a period to each queue of type $r$. When $k\to\infty$, our main result shows that all the policies in this set are equivalent , in the sense that they yield the same mean stationary waiting time, and optimal , in the sense that no other policy having the same aggregate arrival rate to all queues of a given type can do better in minimizing the stationary mean waiting time. This property holds in a strong probabilistic sense. Furthermore, the limiting mean waiting time achieved by our policies is a convex function of the arrival rate in each queue, which facilitates the development of a further optimization aimed at solving the fundamental problem above for large systems.
</p>projecteuclid.org/euclid.ssy/1450879283_20151223090124Wed, 23 Dec 2015 09:01 ESTOn patient flow in hospitals: A data-based queueing-science perspectivehttp://projecteuclid.org/euclid.ssy/1450879284<strong>Mor Armony</strong>, <strong>Shlomo Israelit</strong>, <strong>Avishai Mandelbaum</strong>, <strong>Yariv N. Marmor</strong>, <strong>Yulia Tseytlin</strong>, <strong>Galit B. Yom-Tov</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 5, Number 1, 146--194.</p><p><strong>Abstract:</strong><br/>
Hospitals are complex systems with essential societal benefits and huge mounting costs. These costs are exacerbated by inefficiencies in hospital processes, which are often manifested by congestion and long delays in patient care. Thus, a queueing-network view of patient flow in hospitals is natural for studying and improving its performance. The goal of our research is to explore patient flow data through the lens of a queueing scientist. The means is exploratory data analysis (EDA) in a large Israeli hospital, which reveals important features that are not readily explainable by existing models.
Questions raised by our EDA include: Can a simple (parsimonious) queueing model usefully capture the complex operational reality of the Emergency Department (ED)? What time scales and operational regimes are relevant for modeling patient length of stay in the Internal Wards (IWs)? How do protocols of patient transfer between the ED and the IWs influence patient delay, workload division and fairness? EDA also underscores the importance of an integrative view of hospital units by, for example, relating ED bottlenecks to IW physician protocols. The significance of such questions and our related findings raise the need for novel queueing models and theory, which we present here as research opportunities .
Hospital data, and specifically patient flow data at the level of the individual patient, is increasingly collected but is typically confidential and/or proprietary. We have been fortunate to partner with a hospital that allowed us to open up its data for everyone to access. This enables reproducibility of our findings, through a user-friendly platform that is accessible via the Technion SEELab.
</p>projecteuclid.org/euclid.ssy/1450879284_20151223090124Wed, 23 Dec 2015 09:01 ESTModels with hidden regular variation: Generation and detectionhttp://projecteuclid.org/euclid.ssy/1450879454<strong>Bikramjit Das</strong>, <strong>Sidney I. Resnick</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 5, Number 2, 195--238.</p><p><strong>Abstract:</strong><br/>
We review the notions of multivariate regular variation (MRV) and hidden regular variation (HRV) for distributions of random vectors and then discuss methods for generating models exhibiting both properties concentrating on the non-negative orthant in dimension two. Furthermore we suggest diagnostic techniques that detect these properties in multivariate data and indicate when models exhibiting both MRV and HRV are plausible fits for the data. We illustrate our techniques on simulated data, as well as two real Internet data sets.
</p>projecteuclid.org/euclid.ssy/1450879454_20151223090416Wed, 23 Dec 2015 09:04 ESTTightness of stationary distributions of a flexible-server system in the Halfin-Whitt asymptotic regimehttp://projecteuclid.org/euclid.ssy/1450879455<strong>Alexander L. Stolyar</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 5, Number 2, 239--267.</p><p><strong>Abstract:</strong><br/>
A large-scale flexible service system with two large server pools and two types of customers is considered. Servers in pool 1 can only serve type 1 customers, while server in pool 2 are flexible – they can serve both types 1 and 2. (This is a so-called “N-system.”) The customer arrival processes are Poisson and customer service requirements are independent exponentially distributed. The service rate of a customer depends both on its type and the pool where it is served. A priority service discipline, where type 2 has priority in pool 2, and type 1 prefers pool 1, is considered. We consider the Halfin-Whitt asymptotic regime, where the arrival rate of customers and the number of servers in each pool increase to infinity in proportion to a scaling parameter $n$, while the overall system capacity exceeds its load by $O(\sqrt{n})$.
For this system we prove tightness of diffusion-scaled stationary distributions. Our approach relies on a single common Lyapunov function $G^{(n)}(x)$, depending on parameter $n$ and defined on the entire state space as a functional of the drift-based fluid limits (DFL). Specifically, $G^{(n)}(x)=\int_{0}^{\infty}g(y^{(n)}(t))dt$, where $y^{(n)}(\cdot)$ is the DFL starting at $x$, and $g(\cdot)$ is a “distance” to the origin. ($g(\cdot)$ is same for all $n$). The key part of the analysis is the study of the (first and second) derivatives of the DFLs and function $G^{(n)}(x)$. The approach, as well as many parts of the analysis, are quite generic and may be of independent interest.
</p>projecteuclid.org/euclid.ssy/1450879455_20151223090416Wed, 23 Dec 2015 09:04 ESTOn bid-price controls for network revenue managementhttp://projecteuclid.org/euclid.ssy/1450879456<strong>Barış Ata</strong>, <strong>Mustafa Akan</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 5, Number 2, 268--323.</p><p><strong>Abstract:</strong><br/>
We consider a network revenue management problem and advance its dual formulation. The dual formulation reveals that the (optimal) shadow price of capacity forms a nonnegative martingale. This result is proved under minimal assumptions on network topology and stochastic nature of demand, allowing an arbitrary statistical dependence structure across time and products. Next, we consider a quadratic perturbation of the network revenue management problem and show that a simple (perturbed) bid-price control is optimal for the perturbed problem; and it is $\varepsilon$-optimal for the original network revenue management problem. Finally, we consider a predictable version of this control, where the bid prices used in the current period are last updated in the previous period, and provide an upper bound on its optimality gap in terms of the (quadratic) variation of demand. Using this upper bound we show that there exists a near-optimal such control in the usual case when periods are small compared to the planning horizon provided that either demand or the incremental information arriving during each period is small. We establish the martingale property of the (near) optimal bid prices in both settings. The martingale property can have important implications in practice as it may offer a tool for monitoring the revenue management systems.
</p>projecteuclid.org/euclid.ssy/1450879456_20151223090416Wed, 23 Dec 2015 09:04 ESTSolving the drift control problemhttp://projecteuclid.org/euclid.ssy/1450879457<strong>Melda Ormeci Matoglu</strong>, <strong>John Vande Vate</strong>, <strong>Huizhu Wang</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 5, Number 2, 324--371.</p><p><strong>Abstract:</strong><br/>
We model the problem of managing capacity in a build-to-order environment as a Brownian drift control problem. We formulate a structured linear program that models a practical discretization of the problem and exploit a strong relationship between relative value functions and dual solutions to develop a functional lower bound for the continuous problem from a dual solution to the discrete problem. Refining the discretization proves a functional strong duality for the continuous problem. The linear programming formulation is so badly scaled, however, that solving it is beyond the capabilities of standard solvers. By demonstrating the equivalence between strongly feasible bases and deterministic unichain policies, we combinatorialize the pivoting process and by exploiting the relationship between dual solutions and relative value functions, develop a mechanism for solving the LP without ever computing its coefficients. Finally, we exploit the relationship between relative value functions and dual solutions to develop a scheme analogous to column generation for refining the discretization so as to drive the gap between the discrete approximation and the continuous problem to zero quickly while keeping the LP small. Computational studies show our scheme is much faster than simply solving a regular discretization of the problem both in terms of finding a policy with a low average cost and in terms of providing a lower bound on the optimal average cost.
</p>projecteuclid.org/euclid.ssy/1450879457_20151223090416Wed, 23 Dec 2015 09:04 ESTGiant component in random multipartite graphs with given degree sequenceshttp://projecteuclid.org/euclid.ssy/1450879458<strong>David Gamarnik</strong>, <strong>Sidhant Misra</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 5, Number 2, 372--408.</p><p><strong>Abstract:</strong><br/>
We study the problem of the existence of a giant component in a random multipartite graph. We consider a random multipartite graph with $p$ parts generated according to a given degree sequence $n_{i}^{\mathbf{d}}(n),n\ge1$ which denotes the number of vertices in part $i$ of the multipartite graph with degree given by the vector $\mathbf{d}$ in an $n$-node graph. We assume that the empirical distribution of the degree sequence converges to a limiting probability distribution. Under certain mild regularity assumptions, we characterize the conditions under which, with high probability, there exists a component of linear size. The characterization involves checking whether the Perron-Frobenius norm of the matrix of means of a certain associated edge-biased distribution is greater than unity. We also specify the size of the giant component when it exists. We use the exploration process of Molloy and Reed Molloy and Reed (1995) to analyze the size of components in the random graph. The main challenges arise due to the multidimensionality of the random processes involved which prevents us from directly applying the techniques from the standard unipartite case. In this paper we use techniques from the theory of multidimensional Galton-Watson processes along with Lyapunov function technique to overcome the challenges.
</p>projecteuclid.org/euclid.ssy/1450879458_20151223090416Wed, 23 Dec 2015 09:04 ESTOn queue-size scaling for input-queued switcheshttp://projecteuclid.org/euclid.ssy/1479287404<strong>D. Shah</strong>, <strong>J. N. Tsitsiklis</strong>, <strong>Y. Zhong</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 6, Number 1, 1--25.</p><p><strong>Abstract:</strong><br/>
We study the optimal scaling of the expected total queue size in an $n\times n$ input-queued switch, as a function of the number of ports $n$ and the load factor $\rho$, which has been conjectured to be $\Theta(n/{(}1-\rho{)})$ (cf. [15]). In a recent work [16], the validity of this conjecture has been established for the regime where $1-\rho=O(1/n^{2})$. In this paper, we make further progress in the direction of this conjecture. We provide a new class of scheduling policies under which the expected total queue size scales as $O\big(n^{1.5}(1-\rho)^{-1}\log\big(1/(1-\rho)\big)\big)$ when $1-\rho=O(1/n)$. This is an improvement over the state of the art; for example, for $\rho=1-1/n$ the best known bound was $O(n^{3})$, while ours is $O(n^{2.5}\log n)$.
</p>projecteuclid.org/euclid.ssy/1479287404_20161116041014Wed, 16 Nov 2016 04:10 ESTDynamic scheduling for parallel server systems in heavy traffic: Graphical structure, decoupled workload matrix and some sufficient conditions for solvability of the Brownian control problemhttp://projecteuclid.org/euclid.ssy/1479287405<strong>V. Pesic</strong>, <strong>R. J. Williams</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 6, Number 1, 26--89.</p><p><strong>Abstract:</strong><br/>
We consider a dynamic scheduling problem for parallel server systems. J. M. Harrison has proposed a scheme for using diffusion control problems to approximately solve such control problems for heavily loaded systems. This approach has been very successfully used in the special case when the diffusion control problem can be reduced to an equivalent one for a one-dimensional workload process. However, it remains a challenging open problem to make substantial progress on using Harrison’s scheme when the workload process is more than one-dimensional. Here we present some new structural results concerning the diffusion control problem for parallel server systems with arbitrary workload dimension. Specifically, we prove that a certain server-buffer graph associated with a parallel server system is a forest of trees. We then exploit this graphical structure to prove that there exists a matrix, used to define the workload process, that has a block diagonal-like structure. An important feature of this matrix is that, except when the workload is one-dimensional, this matrix is frequently different from a choice of workload matrix proposed by Harrison. We demonstrate that our workload matrix simplifies the structure of the control problem for the workload process by proving that when the original diffusion control problem has linear holding costs, the equivalent workload formulation also has a linear cost function. We also use this simplification to give sufficient conditions for a certain least control process to be an optimal control for the diffusion control problem with linear holding costs. Under these conditions, we propose a continuous review threshold-type control policy for the original parallel server system that exploits pooling of servers within trees in the server-buffer graph and uses non-basic activities connecting different trees in a critical manner. We call this partial pooling. We conjecture that this threshold policy is asymptotically optimal in the heavy traffic limit. We illustrate the solution of the diffusion control problem and our proposed threshold control policy for a three-buffer, three-server example.
</p>projecteuclid.org/euclid.ssy/1479287405_20161116041014Wed, 16 Nov 2016 04:10 ESTRandomized assignment of jobs to servers in heterogeneous clusters of shared servers for low delayhttp://projecteuclid.org/euclid.ssy/1479287406<strong>Arpan Mukhopadhyay</strong>, <strong>A. Karthik</strong>, <strong>Ravi R. Mazumdar</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 6, Number 1, 90--131.</p><p><strong>Abstract:</strong><br/>
We consider the problem of assignning jobs to servers in a multi-server system consisting of $N$ parallel processor sharing servers, categorized into $M$ ($\ll N$) different types according to their processing capacities or speeds. Jobs of random sizes arrive at the system according to a Poisson process with rate $N\lambda$. Upon each arrival, some servers of each type are sampled uniformly at random. The job is then assigned to one of the sampled servers based on their states. We propose two schemes, which differ in the metric for choosing the destination server for each arriving job. Our aim is to reduce the mean sojourn time of the jobs in the system.
It is shown that the proposed schemes achieve the maximal stability region, without requiring the knowledge of the system parameters. The performance of the system operating under the proposed schemes is analyzed in the limit as $N\to\infty$. This gives rise to a mean field limit. The mean field is shown to have a unique, globally asymptotically stable equilibrium point which approximates the stationary distribution of load at each server. Asymptotic independence among the servers is established using a notion of intra-type exchangeability which generalizes the usual notion of exchangeability. It is further shown that the tail distribution of server occupancies decays doubly exponentially for each server type. Numerical evidence shows that at high load the proposed schemes perform at least as well as other schemes that require more knowledge of the system parameters.
</p>projecteuclid.org/euclid.ssy/1479287406_20161116041014Wed, 16 Nov 2016 04:10 ESTChattering and congestion collapse in an overload switching controlhttp://projecteuclid.org/euclid.ssy/1479287407<strong>Ohad Perry</strong>, <strong>Ward Whitt</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 6, Number 1, 132--210.</p><p><strong>Abstract:</strong><br/>
Routing mechanisms for stochastic networks are often designed to produce state space collapse (SSC) in a heavy-traffic limit, i.e., to confine the limiting process to a lower-dimensional subset of its full state space. In a fluid limit, a control producing asymptotic SSC corresponds to an ideal sliding mode control that forces the fluid trajectories to a lower-dimensional sliding manifold . Within deterministic dynamical systems theory, it is well known that sliding-mode controls can cause the system to chatter back and forth along the sliding manifold due to delays in activation of the control. For the prelimit stochastic system, chattering implies fluid-scaled fluctuations that are larger than typical stochastic fluctuations.
In this paper we show that chattering can occur in the fluid limit of a controlled stochastic network when inappropriate control parameters are used. The model has two large service pools operating under the fixed-queue-ratio with activation and release thresholds (FQR-ART) overload control which we proposed in a recent paper. The FQR-ART control is designed to produce asymptotic SSC by automatically activating sharing (sending some customers from one class to the other service pool) once an overload occurs. We have previously shown that this control is effective and robust, even if the service rates are less for the other shared customers, when the control parameters are chosen properly. We now show that, if the control parameters are not chosen properly, then delays in activating and releasing the control can cause chattering with large oscillations in the fluid limit. In turn, these fluid-scaled fluctuations lead to severe congestion, even when the arrival rates are smaller than the potential total service rate in the system, a phenomenon referred to as congestion collapse . We show that the fluid limit can be a bi-stable switching system possessing a unique nontrivial periodic equilibrium, in addition to a unique stationary point.
</p>projecteuclid.org/euclid.ssy/1479287407_20161116041014Wed, 16 Nov 2016 04:10 ESTHeavy traffic queue length behavior in a switch under the MaxWeight algorithmhttp://projecteuclid.org/euclid.ssy/1479287408<strong>Siva Theja Maguluri</strong>, <strong>R. Srikant</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 6, Number 1, 211--250.</p><p><strong>Abstract:</strong><br/>
We consider a switch operating under the MaxWeight scheduling algorithm, under any traffic pattern such that all the ports are loaded. This system is interesting to study since the queue lengths exhibit a multi-dimensional state-space collapse in the heavy-traffic regime. We use a Lyapunov-type drift technique to characterize the heavy-traffic behavior of the expectation of the sum queue lengths in steady-state, under the assumption that all ports are saturated and all queues receive non-zero traffic. Under these conditions, we show that the heavy-traffic scaled queue length is given by $(1-\frac{1}{2n})||\sigma||^{2}$, where $\sigma$ is the vector of the standard deviations of arrivals to each port in the heavy-traffic limit. In the special case of uniform Bernoulli arrivals, the corresponding formula is given by $(n-\frac{3}{2}+\frac{1}{2n})$. The result shows that the heavy-traffic scaled queue length has optimal scaling with respect to $n,$ thus settling one version of an open conjecture; in fact, it is shown that the heavy-traffic queue length is at most within a factor of two from the optimal. We then consider certain asymptotic regimes where the load of the system scales simultaneously with the number of ports. We show that the MaxWeight algorithm has optimal queue length scaling behavior provided that the arrival rate approaches capacity sufficiently fast.
</p>projecteuclid.org/euclid.ssy/1479287408_20161116041014Wed, 16 Nov 2016 04:10 ESTAsymptotic behavior of a critical fluid model for a processor sharing queue via relative entropyhttp://projecteuclid.org/euclid.ssy/1490148013<strong>Amber L. Puha</strong>, <strong>Ruth J. Williams</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 6, Number 2, 251--300.</p><p><strong>Abstract:</strong><br/>
In this paper, we develop a new approach to studying the asymptotic behavior of fluid model solutions for critically loaded processor sharing queues. For this, we introduce a notion of relative entropy associated with measure-valued fluid model solutions. In contrast to the approach used in [12], which does not readily generalize to networks of processor sharing queues, we expect the approach developed in this paper to be more robust. Indeed, we anticipate that similar notions involving relative entropy may be helpful for understanding the asymptotic behavior of critical fluid model solutions for stochastic networks operating under various resource sharing protocols naturally described by measure-valued processes.
</p>projecteuclid.org/euclid.ssy/1490148013_20170321220020Tue, 21 Mar 2017 22:00 EDTStein’s method for steady-state diffusion approximations: An introduction through the Erlang-A and Erlang-C modelshttp://projecteuclid.org/euclid.ssy/1490148014<strong>Anton Braverman</strong>, <strong>J. G. Dai</strong>, <strong>Jiekun Feng</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 6, Number 2, 301--366.</p><p><strong>Abstract:</strong><br/>
This paper provides an introduction to the Stein method framework in the context of steady-state diffusion approximations. The framework consists of three components: the Poisson equation and gradient bounds, generator coupling, and moment bounds. Working in the setting of the Erlang-A and Erlang-C models, we prove that both Wasserstein and Kolmogorov distances between the stationary distribution of a normalized customer count process, and that of an appropriately defined diffusion process decrease at a rate of $1/\sqrt{R}$, where $R$ is the offered load. Futhermore, these error bounds are universal , valid in any load condition from lightly loaded to heavily loaded.
</p>projecteuclid.org/euclid.ssy/1490148014_20170321220020Tue, 21 Mar 2017 22:00 EDTConvergence properties of weighted particle islands with application to the double bootstrap algorithmhttp://projecteuclid.org/euclid.ssy/1490148015<strong>Pierre Del Moral</strong>, <strong>Eric Moulines</strong>, <strong>Jimmy Olsson</strong>, <strong>Christelle Vergé</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 6, Number 2, 367--419.</p><p><strong>Abstract:</strong><br/>
Particle island models [31] provide a means of parallelization of sequential Monte Carlo methods, and in this paper we present novel convergence results for algorithms of this sort. In particular we establish a central limit theorem—as the number of islands and the common size of the islands tend jointly to infinity—of the double bootstrap algorithm with possibly adaptive selection on the island level. For this purpose we introduce a notion of archipelagos of weighted islands and find conditions under which a set of convergence properties are preserved by different operations on such archipelagos. This theory allows arbitrary compositions of these operations to be straightforwardly analyzed, providing a very flexible framework covering the double bootstrap algorithm as a special case. Finally, we establish the long-term numerical stability of the double bootstrap algorithm by bounding its asymptotic variance under weak and easily checked assumptions satisfied typically for models with non-compact state space.
</p>projecteuclid.org/euclid.ssy/1490148015_20170321220020Tue, 21 Mar 2017 22:00 EDTClearing analysis on phases: Exact limiting probabilities for skip-free, unidirectional, quasi-birth-death processeshttp://projecteuclid.org/euclid.ssy/1490148016<strong>Sherwin Doroudi</strong>, <strong>Brian Fralix</strong>, <strong>Mor Harchol-Balter</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 6, Number 2, 420--458.</p><p><strong>Abstract:</strong><br/>
A variety of problems in computing, service, and manufacturing systems can be modeled via infinite repeating Markov chains with an infinite number of levels and a finite number of phases. Many such chains are quasi-birth-death processes with transitions that are skip-free in level , in that one can only transition between consecutive levels, and unidirectional in phase , in that one can only transition from lower-numbered phases to higher-numbered phases. We present a procedure, which we call Clearing Analysis on Phases (CAP), for determining the limiting probabilities of such Markov chains exactly. The CAP method yields the limiting probability of each state in the repeating portion of the chain as a linear combination of scalar bases raised to a power corresponding to the level of the state. The weights in these linear combinations can be determined by solving a finite system of linear equations.
</p>projecteuclid.org/euclid.ssy/1490148016_20170321220020Tue, 21 Mar 2017 22:00 EDTConstruction of asymptotically optimal control for crisscross network from a free boundary problemhttp://projecteuclid.org/euclid.ssy/1490148017<strong>Amarjit Budhiraja</strong>, <strong>Xin Liu</strong>, <strong>Subhamay Saha</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 6, Number 2, 459--518.</p><p><strong>Abstract:</strong><br/>
An asymptotic framework for optimal control of multiclass stochastic processing networks, using formal diffusion approximations under suitable temporal and spatial scaling, by Brownian control problems (BCP) and their equivalent workload formulations (EWF), has been developed by Harrison (1988). This framework has been implemented in many works for constructing asymptotically optimal control policies for a broad range of stochastic network models. To date all asymptotic optimality results for such networks correspond to settings where the solution of the EWF is a reflected Brownian motion in $\mathbb{R} _{+}$ or a wedge in $\mathbb{R} _{+}^{2}$. In this work we consider a well studied stochastic network which is perhaps the simplest example of a model with more than one dimensional workload process. In the regime considered here, the singular control problem corresponding to the EWF does not have a simple form explicit solution. However, by considering an associated free boundary problem one can give a representation for an optimal controlled process as a two dimensional reflected Brownian motion in a Lipschitz domain whose boundary is determined by the solution of the free boundary problem. Using the form of the optimal solution we propose a sequence of control policies, given in terms of suitable thresholds, for the scaled stochastic network control problems and prove that this sequence of policies is asymptotically optimal. As suggested by the solution of the EWF, the policy we propose requires a server to idle under certain conditions which are specified in terms of thresholds determined from the free boundary.
</p>projecteuclid.org/euclid.ssy/1490148017_20170321220020Tue, 21 Mar 2017 22:00 EDTHeavy-traffic limits for a fork-join networkin the Halfin-Whitt regimehttp://projecteuclid.org/euclid.ssy/1490148018<strong>Hongyuan Lu</strong>, <strong>Guodong Pang</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 6, Number 2, 519--600.</p><p><strong>Abstract:</strong><br/>
We study a fork-join network with a single class of jobs, which are forked into a fixed number of parallel tasks upon arrival to be processed at the corresponding multi-server stations. After service completion, each task will join a buffer associated with the service station waiting for synchronization, called “unsynchronized queue”. The synchronization rule requires that all tasks from the same job must be completed, referred to as “non-exchangeable synchronization”. Once synchronized, jobs will leave the system immediately. Service times of the parallel tasks of each job can be correlated and form a sequence of i.i.d. random vectors with a general continuous joint distribution function. We study the joint dynamics of the queueing and service processes at all stations and the associated unsynchronized queueing processes.
The main mathematical challenge lies in the “resequencing” of arrival orders after service completion at each station. As in Lu and Pang (2015) for the infinite-server fork-join network model, the dynamics of all the aforementioned processes can be represented via a multiparameter sequential empirical process driven by the service vectors for the parallel tasks of each job. We consider the system in the Halfin-Whitt regime, and prove a functional law of large number and a functional central limit theorem for queueing and synchronization processes. In this regime, although the delay for service at each station is asymptotically negligible, the delay for synchronization is of the same order as the service times.
</p>projecteuclid.org/euclid.ssy/1490148018_20170321220020Tue, 21 Mar 2017 22:00 EDTScaling limits for infinite-server systems in a random environmenthttp://projecteuclid.org/euclid.ssy/1495785616<strong>Mariska Heemskerk</strong>, <strong>Johan van Leeuwaarden</strong>, <strong>Michel Mandjes</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 7, Number 1, 1--31.</p><p><strong>Abstract:</strong><br/>
This paper studies the effect of an overdispersed arrival process on the performance of an infinite-server system. In our setup, a random environment is modeled by drawing an arrival rate $\Lambda$ from a given distribution every $\Delta$ time units, yielding an i.i.d. sequence of arrival rates $\Lambda_{1},\Lambda_{2},\ldots$. Applying a martingale central limit theorem, we obtain a functional central limit theorem for the scaled queue length process. We proceed to large deviations and derive the logarithmic asymptotics of the queue length’s tail probabilities. As it turns out, in a rapidly changing environment (i.e., $\Delta$ is small relative to $\Lambda$) the overdispersion of the arrival process hardly affects system behavior, whereas in a slowly changing random environment it is fundamentally different; this general finding applies to both the central limit and the large deviations regime. We extend our results to the setting where each arrival creates a job in multiple infinite-server queues.
</p>projecteuclid.org/euclid.ssy/1495785616_20170526040018Fri, 26 May 2017 04:00 EDTAsymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approachhttp://projecteuclid.org/euclid.ssy/1495785617<strong>Sandro Franceschi</strong>, <strong>Irina Kourkova</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 7, Number 1, 32--94.</p><p><strong>Abstract:</strong><br/>
Brownian motion in $\mathbf{R}_{+}^{2}$ with covariance matrix $\Sigma$ and drift $\mu$ in the interior and reflection matrix $R$ from the axes is considered. The asymptotic expansion of the stationary distribution density along all paths in $\mathbf{R}_{+}^{2}$ is found and its main term is identified depending on parameters $(\Sigma,\mu,R)$. For this purpose the analytic approach of Fayolle, Iasnogorodski and Malyshev in [12] and [36], restricted essentially up to now to discrete random walks in $\mathbf{Z}_{+}^{2}$ with jumps to the nearest-neighbors in the interior is developed in this article for diffusion processes on $\mathbf{R}_{+}^{2}$ with reflections on the axes.
</p>projecteuclid.org/euclid.ssy/1495785617_20170526040018Fri, 26 May 2017 04:00 EDTHeavy-traffic limit for the initial content processhttp://projecteuclid.org/euclid.ssy/1495785618<strong>A. Korhan Aras</strong>, <strong>Yunan Liu</strong>, <strong>Ward Whitt</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 7, Number 1, 95--142.</p><p><strong>Abstract:</strong><br/>
To understand the performance of a queueing system, it can be useful to focus on the evolution of the content that is initially in service at some time. That necessarily will be the case in service systems that provide service during normal working hours each day, with the system shutting down at some time, except that all customers already in service at the termination time are allowed to complete their service. Also, for infinite-server queues, it is often fruitful to partition the content into the initial content and the new input, because these can be analyzed separately. With i.i.d service times having a non-exponential distribution, the state of the initial content can be described by specifying the elapsed service times of the remaining initial customers. That initial content process is then a Markov process. This paper establishes a many-server heavy-traffic (MSHT) functional central limit theorem (FCLT) for the initial content process in the space $\mathbb{D}_{\mathbb{D}}$, assuming a FCLT for the initial age process, with the number of customers initially in service growing in the limit. The proof applies a symmetrization lemma from the literature on empirical processes to address a technical challenge: For each time, including time $0$, the conditional remaining service times, given the ages, are mutually independent but in general not identically distributed.
</p>projecteuclid.org/euclid.ssy/1495785618_20170526040018Fri, 26 May 2017 04:00 EDTHeavy traffic approximation for the stationary distribution of a generalized Jackson network: The BAR approachhttp://projecteuclid.org/euclid.ssy/1495785619<strong>Anton Braverman</strong>, <strong>J. G. Dai</strong>, <strong>Masakiyo Miyazawa</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 7, Number 1, 143--196.</p><p><strong>Abstract:</strong><br/>
In the seminal paper of Gamarnik and Zeevi [17], the authors justify the steady-state diffusion approximation of a generalized Jackson network (GJN) in heavy traffic. Their approach involves the so-called limit interchange argument, which has since become a popular tool employed by many others who study diffusion approximations. In this paper we illustrate a novel approach by using it to justify the steady-state approximation of a GJN in heavy traffic. Our approach involves working directly with the basic adjoint relationship (BAR), an integral equation that characterizes the stationary distribution of a Markov process. As we will show, the BAR approach is a more natural choice than the limit interchange approach for justifying steady-state approximations, and can potentially be applied to the study of other stochastic processing networks such as multiclass queueing networks.
</p>projecteuclid.org/euclid.ssy/1495785619_20170526040018Fri, 26 May 2017 04:00 EDTWaves in a spatial queuehttp://projecteuclid.org/euclid.ssy/1495785620<strong>David Aldous</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 7, Number 1, 197--236.</p><p><strong>Abstract:</strong><br/>
Envisaging a physical queue of humans, we model a long queue by a continuous-space model in which, when a customer moves forward, they stop a random distance behind the previous customer, but do not move at all if their distance behind the previous customer is below a threshold. The latter assumption leads to “waves” of motion in which only some random number $W$ of customers move. We prove that $\mathbb{P}(W>k)$ decreases as order $k^{-1/2}$; in other words, for large $k$ the $k$’th customer moves on average only once every order $k^{1/2}$ service times. A more refined analysis relies on a non-obvious asymptotic relation to the coalescing Brownian motion process; we give a careful outline of such an analysis without attending to all the technical details.
</p>projecteuclid.org/euclid.ssy/1495785620_20170526040018Fri, 26 May 2017 04:00 EDTA Blood Bank Model with Perishable Blood and Demand Impatiencehttps://projecteuclid.org/euclid.ssy/1519441213<strong>Shaul K. Bar-Lev</strong>, <strong>Onno Boxma</strong>, <strong>Britt Mathijsen</strong>, <strong>David Perry</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 7, Number 2, 237--262.</p><p><strong>Abstract:</strong><br/>
We consider a stochastic model for a blood bank, in which amounts of blood are offered and demanded according to independent compound Poisson processes. Blood is perishable, i.e., blood can only be kept in storage for a limited amount of time. Furthermore, demand for blood is impatient, i.e., a demand for blood may be canceled if it cannot be satisfied soon enough. For a range of perishability functions and demand impatience functions, we derive the steady-state distributions of the amount of blood kept in storage, and of the amount of demand for blood (at any point in time, at most one of these quantities is positive). Under certain conditions we also obtain the fluid and diffusion limits of the blood inventory process, showing in particular that the diffusion limit process is an Ornstein-Uhlenbeck process.
</p>projecteuclid.org/euclid.ssy/1519441213_20180223220016Fri, 23 Feb 2018 22:00 ESTAn Ergodic Control Problem for Many-Server Multiclass Queueing Systems with Cross-Trained Servershttps://projecteuclid.org/euclid.ssy/1519441214<strong>Anup Biswas</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 7, Number 2, 263--288.</p><p><strong>Abstract:</strong><br/>
A Markovian queueing network is considered with d independent customer classes and d server pools in Halfin–Whitt regime. Class i customers has priority for service in pool i for i = 1, …, d , and may access some other pool if the pool has an idle server and all the servers in pool i are busy. We formulate an ergodic control problem where the running cost is given by a non-negative convex function with polynomial growth. We show that the limiting controlled diffusion is modelled by an action space which depends on the state variable. We provide a complete analysis for the limiting ergodic control problem and establish asymptotic convergence of the value functions for the queueing model.
</p>projecteuclid.org/euclid.ssy/1519441214_20180223220016Fri, 23 Feb 2018 22:00 ESTDetecting Markov Chain Instability: A Monte Carlo Approachhttps://projecteuclid.org/euclid.ssy/1519441215<strong>M. Mandjes</strong>, <strong>B. Patch</strong>, <strong>N. S. Walton</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 7, Number 2, 289--314.</p><p><strong>Abstract:</strong><br/> We devise a Monte Carlo based method for detecting whether a non-negative Markov chain is stable for a given set of parameter values. More precisely, for a given subset of the parameter space, we develop an algorithm that is capable of deciding whether the set has a subset of positive Lebesgue measure for which the Markov chain is unstable. The approach is based on a variant of simulated annealing, and consequently only mild assumptions are needed to obtain performance guarantees. The theoretical underpinnings of our algorithm are based on a result stating that the stability of a set of parameters can be phrased in terms of the stability of a single Markov chain that searches the set for unstable parameters. Our framework leads to a procedure that is capable of performing statistically rigorous tests for instability, which has been extensively tested using several examples of standard and non-standard queueing networks. </p>projecteuclid.org/euclid.ssy/1519441215_20180223220016Fri, 23 Feb 2018 22:00 ESTOptimal Admission Control for Many-Server Systems with QED-Driven Revenueshttps://projecteuclid.org/euclid.ssy/1519441216<strong>Jaron Sanders</strong>, <strong>S. C. Borst</strong>, <strong>A. J. E. M. Janssen</strong>, <strong>J. S. H. van Leeuwaarden</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 7, Number 2, 315--341.</p><p><strong>Abstract:</strong><br/>
We consider Markovian many-server systems with admission control operating in a Quality-and-Efficiency-Driven (QED) regime, where the relative utilization approaches unity while the number of servers grows large, providing natural Economies-of-Scale. In order to determine the optimal admission control policy, we adopt a revenue maximization framework, and suppose that the revenue rate attains a maximum when no customers are waiting and no servers are idling. When the revenue function scales properly with the system size, we show that a nondegenerate optimization problem arises in the limit. Detailed analysis demonstrates that the revenue is maximized by nontrivial policies that bar customers from entering when the queue length exceeds a certain threshold of the order of the typical square-root level variation in the system occupancy. We identify a fundamental equation characterizing the optimal threshold, which we extensively leverage to provide broadly applicable upper/lower bounds for the optimal threshold, establish its monotonicity, and examine its asymptotic behavior, all for general revenue structures. For linear and exponential revenue structures, we present explicit expressions for the optimal threshold.
</p>projecteuclid.org/euclid.ssy/1519441216_20180223220016Fri, 23 Feb 2018 22:00 ESTScaling Limit of a Limit Order Book Model via the Regenerative Characterization of Lévy Treeshttps://projecteuclid.org/euclid.ssy/1519441217<strong>Peter Lakner</strong>, <strong>Josh Reed</strong>, <strong>Florian Simatos</strong>. <p><strong>Source: </strong>Stochastic Systems, Volume 7, Number 2, 342--373.</p><p><strong>Abstract:</strong><br/> We consider the following Markovian dynamic on point processes: at constant rate and with equal probability, either the rightmost atom of the current configuration is removed, or a new atom is added at a random distance from the rightmost atom. Interpreting atoms as limit buy orders, this process was introduced by Lakner et al. [Lakner et al. (2016) High frequency asymptotics for the limit order book. Mark. Microstructure Liq. 2:1650004 [83 pages]] to model a one-sided limit order book. We consider this model in the regime where the total number of orders converges to a reflected Brownian motion, and complement the results of Lakner et al. [Lakner P, Reed J, Stoikov S (2016) High frequency asymptotics for the limit order book. Mark. Microstructure Liq. 2:1650004 [83 pages]] by showing that, in the case where the mean displacement at which a new order is added is positive, the measure-valued process describing the whole limit order book converges to a simple functional of this reflected Brownian motion. Our results make it possible to derive useful and explicit approximations on various quantities of interest such as the depth or the total value of the book. Our approach leverages an unexpected connection with Lévy trees. More precisely, the cornerstone of our approach is the regenerative characterization of Lévy trees due to Weill [Weill M (2007) Regenerative real trees. Ann. Probab. 35:2091–2121. MR2353384 (2008j:60205)], which provides an elegant proof strategy which we unfold. </p>projecteuclid.org/euclid.ssy/1519441217_20180223220016Fri, 23 Feb 2018 22:00 EST