ON THE BEHAVIOR OF LIFO PREEMPTIVE RESUME QUEUES IN HEAVY TRAFFIC

This paper studies heavy tra(cid:14)c behavior of a G/G/1 last-in-(cid:12)rst-out (LIFO) preemptive resume queue, by extending the techniques developed in Limic (1999). The queue length process exhibits a perhaps unexpected heavy tra(cid:14)c behavior. The di(cid:11)usion limit depends on the type of arrivals (and services) in a fairly intricate way, related to the Wiener-Hopf factorization for random walks.


Introduction
Customers arrive to a single-server queue according to a renewal process with inter-arrival time distribution G, each customer requests service time with distribution function F , independently of other customers. Distributions F and G are concentrated on (0, ∞), and we assume they have finite means m and 1/λ, respectively. The server devotes all of its service potential to the last customer to have arrived. Moreover, at the moment of each new arrival the server switches instantaneously from serving the current customer c (if any) to the newest customerc. Customer c stays waiting in queue and only afterc is served completely and exits the queue does the service of c resume. The server is busy whenever the queue is non-empty, which is usually referred to as a non-idling or work conserving property. The queueing process generated by this mechanism, is a single-class G/G/1 last-in-first-out preemptive resume queue. We prefer to shorten the name to G/G/1 LIFO queue. Special cases are M/G/1 LIFO queues, where the inter-arrival distribution G is exponential (rate λ). Suppose a customer arrives to the queue at time t and requests an amount v of service time. If we let u(s), s ≥ t be its total amount of time in service by time s, the residual service time of this customer at time s is v − u(s) ≥ 0. Denote by (A(t), t ≥ 0) the renewal process of arrivals, by Z(t) the queue length at time t, i.e., the number of individuals in queue at time t, and by W (t) the (immediate) workload of the queue at time t, i.e., the total amount of work still required by customers present in the system at time t (measured in units of server time). Hence, the workload equals the total sum of all the residual service times of customers in queue. The parameter ρ = mλ, called the traffic intensity of the queue, is the average amount of work arriving per unit time. It is a well-known (and easy, cf. section 2.1) fact that the workload process does not vary over work conserving service disciplines. In particular, the workload process (W (t), t ≥ 0) is the same for the first-in-first-out (FIFO) queue, where the customers are served in the order of their arrival. For a M/G/1 LIFO queue, the workload (W (t), t ≥ 0) is a Markov process and it is positive recurrent, null-recurrent, and transient whenever ρ < 1, ρ = 1 and ρ > 1 respectively. For a G/G/1 LIFO queue, the workload is not Markov anymore, but using a random walk comparison, it is easy to see that W returns to 0 infinitely often iff ρ ≤ 1, and the expected time until return is finite iff ρ < 1. From a practical point of view it is desirable to "keep the server busy" most of the time without getting it overwhelmed with work. This corresponds to the situation ρ = 1 − ε for some small ε > 0, and as ε 0 the queue approaches heavy traffic.
A recent work (Limic [12]) describes the heavy traffic behavior for the M/G/1 LIFO queue under the usual second moment assumptions on the service distribution F . The analysis in [12] is based on the following observation. The state of a M/G/1 LIFO queue at any time t (i.e., the list of residual service times) is encoded via a finite-measure-valued Markov process q t , called the the RES-measure process. It is defined in terms of the queue length and the "future minimum" of the load (cf. section 2.4). An analogue of q t is the exploration process introduced in Le Gall and Le Jan [8,7]. The above encoding carries over to the present setting.
The goal of this paper is to extend the techniques of [12] in order to study heavy traffic behavior of G/G/1 LIFO queue under the usual heavy traffic assumptions. The LIFO preemptive resume service discipline induces an essentially different heavy traffic (diffusion scale) behavior from those induced by FIFO service discipline in that the limit (or limit points) depends on the type of arrivals (and services) in a more complicated way than via asymptotic behavior of the first two moments (cf. section 4). However, the "state-space-collapse" property (first discovered by Reiman [14] and common in FIFO-type setting, cf. Bramson [3] and Williams [16]), still holds in a weaker form: under more stringent assumptions, the queue length becomes a multiple of the workload in heavy traffic. Amber Puha and Ruth Williams (personal communication) study the fluid (law of large numbers) scale behavior of the processor sharing queue, where the server simultaneously serves all the customers in queue. The state-space-collapse suggested in their work is analogous to the one for LIFO queues, in that the fluid limit depends on the finer properties of the inter-arrival and service time distributions.
Heavy traffic analysis of general multiclass G/G/1 LIFO preemptive resume queues with Markovian feedback, where the (global) arrival process is not a renewal process (see Reiman [15] and Dai and Kurtz [4] for the FIFO analogues) seems to be beyond the scope of techniques presented here.
The paper is organized as follows. Section 2 describes the processes of interest, and some of their properties. Section 3 is the heavy traffic analysis for a sequence of identically distributed critical (ρ = 1) G/G/1 LIFO queues. Section 4 gives an example, and discusses related complexity issues in analyzing non-identical near critical G/G/1 LIFO queues approaching heavy traffic.
For any two numbers x, y, let x + , x ∧ y, and x ∨ y denote the positive part of x, the minimum, and the maximum of x and y, respectively. We identify H(t) with H t whenever H is a stochastic process.

LIFO queue and related processes
In this section we introduce several processes related to LIFO queues, and mention some important relations. Consider a G/G/1 LIFO (preemptive-resume) queue as in section 1.
Assume that the queue is empty at time t = 0.

The load and the workload
Let (v i : i ≥ 1) be the service times (i.i.d. random variables with distribution F ) requested by the customers in the order of their arrival. Let (u i : i ≥ 1) be the i.i.d. inter-arrival times of customers, where u 1 The load X t and the workload W t of the queue at time t are given by where A(t) = sup{j : j i=1 u j ≤ t} equals the number of customers that arrived to the queue in the time interval [0, t], and I t = inf s≤t X s . Note that the process −I t = − inf s≤t X s is the cumulative idletime of the server by time t, that is, the total time |{s ≤ t : W s = 0}| with no customer in queue. The workload process W is the load process X, reflected above its past infimum. It is easy to see that (1) agrees with the notion of workload in section 1. The excursions of X above the past infimum, or equivalently, the excursions of W above 0 correspond to the busy cycles of the queue.

The queue length and time-reversal
At time γ s the customer that arrived at time s exits the queue, in the meantime its service might be interrupted several times due to jumps of X, that is, arrivals of new customers. We identify the actual set of times when this customer is in service with the set A s = {u ∈ [s, γ s ] : inf t∈[s,u] X t ≥ X u }, indicated in bold on the time axis in the figure. The "gaps" in A s correspond to services of the "intermittent" customers. The customer who arrived (jumped) at time s will still be in queue at time t > s if and only if γ s > t, that is, (as it happens for s and t in the figure).

Electronic Communications in Probability
The difference (inf u∈[s,t] X u − X s− ) + is its residual service time at t. Therefore, the queue length process Z t = Z(t) satisfies Let I t s = inf u∈[s,t] X u be the future infimum process (dotted line in figure 1) of X up to time t. The jumps of I t · may occur only at the times s < t at which customers arrive, and the jump sizes (inf u∈[s,t] X u − X s− ) + are the residual service times at time t of the corresponding customers. The following observation will be important for deriving the queue length heavy traffic approximation. If we fix any time t and time-reverse the load X from t back to 0 (or equivalently, rotate the figure 1 about the origin by 180 degrees), the future infimum I t · "gets mapped" onto the (past) supremum process of the time-reversed load process. In particular, the queue length Z t which equals the number of jumps of the future infimum by (3), also equals the number of jumps of the time-reversed supremum process occurring in [0, t]. In symbols, Note that the time-reversed load process ( X t s , s ∈ [0, t]) does not have the same law as (X s , s ∈ [0, t]), unless the arrival process is Poisson.

Intrinsic branching
Suppose we call a customer who arrives at time t a descendant of a customer that arrived at time s if the latter is still in queue at time t, that is, if (2) holds. This procedure determines a one-to-one correspondence between the busy cycles of the queue and a sequence of independent identically distributed random trees. Any customer either finds the queue empty upon arrival, in which case it becomes a progenitor (or root), or finds the queue non-empty, in which case it becomes a child of the customer being served immediately prior to its arrival. Figure 2 shows a part of the tree induced by the first busy cycle from figure 1.

Figure 2
In the special case of Poisson arrivals (M/G/1), the above trees are clearly Galton-Watson and their offspring distribution can be easily expressed in terms of service distribution F . In the case of renewal arrivals, the Galton-Watson property is preserved. Perhaps the best way to verify this is by considering the children of a particular vertex, denote them by c s1 , c s2 , . . . , c sK , where s 1 < s 2 . . . < s K are the arrival times of the corresponding customers to the queue. Consider the subtrees spanned by c si and its descendents, for i = 1, . . . , K. It suffices to show that, conditionally on K, the K subtrees above are mutually independent, and have the law of the whole tree (corresponding to a complete busy cycle). This is all easily verified from the fact that the ith subtree is determined by the "excursion" The offspring distribution now depends on the inter-arrival and service distributions of the queue, though finding an explicit formula seems to be difficult. We return to this very useful branching characterization in the heavy traffic analysis, section 3.3. The relation between queueing and branching goes back to Kendall [10], and the relation between excursions of random walks and branching goes back to Harris [9].

The RES-measure
One can think of a LIFO queue as a continuous-time process with values in the state space of finite lists of arbitrary length. At each time t, the state of the queue is the list of residual service times for all queued customers ordered by their arrival times. It is convenient to encode the above list via the RES-measure process (q t , t ≥ 0) that takes values in the space M f (R + ) of finite measures (with finite support) on [0, ∞). The queue length can be recovered from q via where Supp(µ) denotes the closed support of µ. Moreover, the list of residual service times at any time t equals (q t (1), q t (2), . . . , q t (Z t )), the list of masses of atoms of q t . The workload is then given by Finally, the process q t is defined by Here µ, ϕ stands for [0,∞) ϕ dµ, where ϕ : R + → R is a continuous function with bounded support, and is the number of individuals in queue at time s that will still be in queue at time t. As in (4), we can express Z t s via time-reversal as Note that the integrals in (6) are in fact finite sums, and the second equality is due to a simple fact Process Z t s is clearly non-decreasing in s for each fixed t. We prefer integrals to sums in (6), since the heavy traffic limit Theorem 3 involves the convergence of rescaled q's to a limit of the same form.

Heavy traffic
Consider a family of G/G/1 queues, indexed by r, with inter-arrival time distribution function G r , and service time distribution function F r . Let F r have finite mean m r , and let G r have finite mean 1/λ r . Denote by A r (·), W r (·), q r (·) and Z r (·) the corresponding arrival, workload, RES-mea-sure, and queue length processes, respectively. We assume that for each r, the queue is empty at time 0 (W r (0) = 0), so the notation of previous sections applies. In particular, rewrite equation (1) as where −I r (t) = − inf s≤t X r (s) is the idle time.

Asymptotics for the load and the workload
The first heavy traffic assumptions are Suppose for a moment that the arrival processes are Poisson (M/G/1 setting) with rate λ r . Assume moreover that for each r the service times have finite second moment β r , and LetX Then an easy application (cf. [12]) of the functional CLT and the continuity mapping theorem shows the convergence ofX r andŴ r in distribution to Brownian motion X, and reflected Brownian motion W = X − I, respectively, where X has drift −c and variance σ 2 = β/m, and I t = inf s≤t X s . For the case of renewal arrivals, assume in addition the second moments η r of the inter-arrival times are finite, and moreover Then the above statement (and the proof) of convergence for the load and the workload processes continues to hold, the only difference being that the limiting variance σ 2 is given by For a sequence of M/G/1 LIFO queues under assumptions (10)- (13), it was shown in [12], that the load, the workload, the RES-measure and the queue length processes, simultaneously satisfy (X r ,Ŵ r ,q r ,Ẑ r ) ⇒ (X, W, q, Z) .
Here " " indicates appropriate rescaling, and ⇒ denotes convergence in distribution with respect to the topology on the corresponding Skorokhod space. The processes X and W are those from the previous paragraph, and the processes q and Z are defined by Z = 2 where I t s = inf u∈[s,t] X u . In the next section we show that, in the case of renewal arrivals, the queue lengthẐ r converges (under much more stringent assumptions) to a limit of the form Z = αW , where the scaling constant α depends on the inter-arrival and service distributions in a fairly intricate way, related to the Wiener-Hopf factorization for random walks (cf. Feller [6]).

Asymptotics for RES-measure and queue length
Consider a sequence of identical (in distribution) G/G/1 LIFO queues with corresponding service and arrival distributions F and G, having finite means m and 1/λ, and finite second moments β and η, respectively. The assumption (11) translates to mλ = 1 and c = 0 , and the moment assumptions (10,12,13,14,15) are automatically satisfied. LetX r (t) = r −1/2 X r (rt) andŴ r (t) = r −1/2 W r (rt) be as before. Since (X r ,Ŵ r ) ⇒ (X, W ), by the Skorokhod representation theorem we may assume that almost surely in the Skorokhod space D R 2 [0, ∞). Rescale the queue length and infimum processes accordingly byẐ where I t (s) = I t s = inf u∈[s,t] X u . Letq r be a measure-valued process defined in analogy to (6) via q r t , ϕ := Let X t denote the time-reversed Brownian motion X, Proof. A suitable modification of the proof of [12], Lemma 3.2.2 yields the result. Fix t > 0 and a finite subdivision 0 ≤ s 1 < s 2 < . . . < s k = t on [0, t]. Identity (8) implieŝ is the supremum process of X rt,r = r −1/2 (X r (rt)−X r ((rt− rs)−)), the rescaled and time-reversed X r . Consider the time-reversed process X rt,r (s) = X r (rt) − X r ((rt − s)−), 0 ≤ s < rt. Denote by M r t (z) the number of jumps of X rt,r above its past maximum in the interval [0, rz]. Note that where p → denotes convergence in probability. Define Then by (21) we get The lemma follows from (23), sinceẐ t,r (·) is non-decreasing for each r and t, and Z t (·) is continuous and non-decreasing for each t.
In order to show (21), it will be convenient to consider an "extension" process ( X rt,t (s), s ≥ 0) of ( X rt,t (s), 0 ≤ s ≤ rt), defined in the following way. Independently of the filtration generated by X r , take a sequence {u −i , i ≥ 1} of i.i.d. random variables with distribution G, and a sequence {v −i , i ≥ 0} of i.i.d. random variables with distribution F . Define X rt,r (rt) = X rt,r (rt−) + v 0 and moreover, X rt,r (rt + z) = X rt,r (rt) + Then the extended process X rt,r decreases deterministically at rate 1 in between successive jumps. The time of the first jump has distribution G 1 (typically = G), and all other inter-jump times are i.i.d. random variables with distribution G. The sizes of all jumps are i.i.d. random variables with distribution F . Let S rt,r be the supremum process of the extended X rt,t . Denote by T r,t 1 < T r,t 2 < . . . the successive increase (jump) times of S rt,r . Denote the corresponding sizes of the overshoots by J r,t 1 , J r,t 2 , . . ., so that J r,t i = X rt,r (T r,t i ) − S rt,r (T r,t i −). If the arrivals are not Poisson, the time-reversal influences the distributions of T r,t 1 and J r,t 1 in some fairly complicated way. However, it is easily seen that the renewal property of arrivals implies that the subsequent overshoots J r,t i , i ≥ 2 are independent and identically distributed random variables. Here it is convenient to have X rt,r defined for all s ≥ 0. In fact, we only need to consider overshoots that happened by (reversed) time rz ≤ rt. Note that and the convergence in (18) implies S t,r (z) → S t (z) a.s., as r → ∞. The number of overshoots M r t (z) was defined only for z < t, but if we let M r t (t) = M r t (t−), equation (24) will also hold for z = t. Since for each r, we have F r = F and G r = G, the distribution of J r,t 2 does not depend on r. Similarly, the distribution of J r,t 2 does not depend on t either. Let The remark after Corollary 2 gives an expression for α, in particular, α ∈ (0, ∞). It is easy to see that M r t (z) → ∞ a.s. as r → ∞, therefore (24) and a law of large numbers yield Then it is easy to see that the overshoots (starting from the second one) in the proof of Lemma 1 are the ascending ladder heights of (S n , n ≥ 1). By [6], Theorem XVIII.5.1 It is hard to determine the exact value of α(F, G) in practice, except when F or G are exponential (or related) distributions (cf. Prabhu [13], and section 4). Let q be as in (17) where Z t is as in Corollary 2, and I t s = inf u∈[s,t] X u .
Again, we extend the proof of the corresponding result in [12] to the present setting. The renewal arrivals require extra care, however, since F r = F and G r = G for all r, the tree estimates in the proof of Proposition 5 below simplify a great deal (compare to [12] by continuity of Z (cf. [5] p.122). Recall that for each t the process (Z t (s), 0 ≤ s ≤ t) given by (22) is continuous, moreover the processes The finite dimensional distributions ofẐ r are converging to those of Z due to Corollary 2.
So it suffices to show the tightness ofẐ r , r ≥ 1 with respect to the Skorokhod topology on The idea is to useẐ r ti−θ ≈Ẑ ti,r (t i − θ) ≈Ẑ ti,r (t i ) =Ẑ r ti ≈ Z ti for small θ, and exploit the monotonicity ofẐ t,r s and Z t s in s. Let F r t be the filtration generated byX r . Observe that, for each r,Ẑ ti,r where (Ẑ r,i u , u ∈ [0, T/n]) has the same law as (Ẑ r u , u ∈ [0, T/n]), and is independent of F r ti−1 . The first inequality in (28) is the monotonicity of Z t,r s in s, the second inequality trivially follows from the interpretation ofẐ ti,r t as the (rescaled) number of individuals in queue at time rt whose service will not have been completed by time rt i . For the last inequality in (28) note that the number of customers that arrive to the queue in the time interval [rt i−1 , rt], and do not exit by time rt, can be bounded from above by 1 + Z * (rt−τ ) + where Z * d = Z, and τ ≥ rt i−1 is the first renewal (arrival) time after rt i−1 . Assume the following Proposition 5 For any fixed ε, η > 0 and any integer n 1 , there exist n ≥ n 1 and r 1 ≥ 1 such that sup whereẐ r,i u are defined in (28). The rest of the proof is the same as in [12]: for ε, η > 0 and T fixed as above, find n 1 large enough so that (26, 27) are satisfied for all n ≥ n 1 . Then find n ≥ n 1 and r 1 so that (29) holds. By Corollary 2, Lemma 1 and (26,27)

Proof of Proposition 5
Recall the branching interpretation for the queue length from section 2.3. Each busy cycle of the queue corresponds to an excursion of the load (workload) process, and yields a Galton-Watson tree T of customers who entered (and exited) the queue during this busy cycle. The generation of a vertex in T is its distance from the root, so the root belongs to generation 0, its children to generation 1, their children to generation 2, etc. Let |T | denote the total size (number of vertices) of T , and let ht(T ) denote the height (the maximal generation) of T , respectively. A customer that arrives at time s, creates a new vertex ς in the corresponding tree. If the queue was empty immediately before the arrival (Z(s−) = 0) then ς becomes the root, otherwise ς becomes a child of the customer whose service was interrupted, in both cases the generation of ς in T equals Z(s−) = Z(s) − 1. For the M/G/1 queue, where the customers arrive as Poisson (rate λ) process, it is easy to see that, given v, the offspring distribution for the trees above is Poisson (rate vλ), that is As mentioned earlier, in the general G/G/1 case, the exact offspring distribution Ξ seems difficult to obtain. Since the excursions of the load (workload) have finite length with probability 1, the corresponding trees have finite size (therefore height), and this is equivalent to Eξ ≤ 1 (e.g. [2]). In fact, it is intuitively obvious that Eξ = 1 which again seems to be tricky to verify via a direct calculation. An indirect way is to note that any increase in service time (e.g. scale by 1 + υ, υ > 0) will induce divergence of the load process to +∞, or equivalently, the corresponding trees will have mean offspring Eξ υ > 1. By uniform integrability, one then argues that Eξ υ → Eξ 0 as υ → 0, but Eξ 0 = Eξ ≤ 1, so it must be Eξ = 1. The variance σ 2 ξ of ξ is finite as well. A crude bound on σ ξ can be obtained in the following way. Recall the set A s from section 2 (and Figure 1 where w 1 , . . . , w k are independent and identically distributed random variables. By truncating w's (from above) if necessary, we may assume 0 < E(w 1 ) < ∞, which turns (32) into an upper bound for P (ξ ≥ k). Therefore, and the first sum is finite due to a large deviation principle, while the second sum is finite due to Ev 2 < ∞. Let T r i,j , 1 ≤ j ≤ M r i be the Galton-Watson trees of the busy cycles started after time 0 and completed before time rT /n, and let T r i be the tree of the busy cycle containing the customer present in service at time rT /n. If the queue is empty at time rT /n, set T r i = ∅ to be the empty tree with ht(∅) = 0. The maximal queue-length sup u∈[0,T /n] Z r,i ru in the interval [0, T/n] is dominated by max 1≤j≤M r i ht(T r i,j ) ∨ ht(T r i ) + 1, the maximal height of all trees (of busy cycles) started in [0, rT/n]. So the proposition will follow from Note that, since F r = F and G r = G, all trees T r i,j , T r i , i, j ≥ 1, have the same critical (mean Eξ = 1) offspring distribution Ξ. Kolchin Aldous, [1] Proposition 24, gives the following estimate for the joint height and total size distribution of the same tree: where Recall M r i is the number of trees corresponding to the completed busy cycles of Z i,r , and let | be the total number of vertices for these trees.

Lemma 6
For any fixed n ≥ 1, there exists r 3 ≥ 1 such that

Lemma 7
There is a constant κ < ∞ such that, for any fixed n 1 ≥ 1 there exist n ≥ n 1 and As in the corresponding proof in [12], use the above lemmas together with (35) to conclude that for any such n and r ≥ r 3 , Similarly, by (34), for any fixed n. The above estimates imply (33). It remains to prove technical Lemmas 6 and 7. Lemma 6 is easy (by Blackwell's renewal theorem) since N r i is the number of renewal arrivals with rate λ in the interval [0, rT/n]. For Lemma 7, let τ x = inf{s ≥ 0 : I s ≤ −x}, where I t = inf 0≤s≤t X s , and X is the load process of the queue. Let M x = #{s ∈ [0, τ x ] : X s− < X s and W s = Z s = 0} be the number of customers arriving to empty queue during the interval [0, τ x ]. So M x is the number of busy cycles (i.e. the number of trees) started in the interval [0, τ x ]. Now consider the infimum processes I r,i t = inf 0≤s≤t X r,i s , and the corresponding τ r,i x := inf{s ≥ 0 :Î r,i s ≤ −x}, with the obvious notation. Since the asymptotic load X is a Brownian motion, it is easy to see that we can find n ≥ n 1 and r 3 large enough so that On the complement of {min 1≤i≤n τ r,i By [13], Lemma I.6 (p.42) and Theorem A2, the corresponding characteristic functions of the overshoot (ascending ladder height) is where ξ j 1 > ξ j 2 are the two roots of the equation such that |ξ j i | ≤ 1, i, j = 1, 2. Of course, ξ 1 1 = ξ 2 1 = 1, and it is easy to check (e.g. through Mathematica) that ξ 1 2 ≈ −0.309208 < ξ 2 2 ≈ −0.306625. From (40) we see that α(F 1 , G 1 ) −1 = ES 1 = 1 − ξ 1 2 = 1 − ξ 1 2 = α(F 2 , G 2 ) −1 . This demonstrates that the natural first and second moment assumptions (10)- (15) are not sufficient for the joint convergence (16), when the inter-arrival times have general distribution. Namely, consider a sequence of G/G/1 LIFO queues, indexed by positive-integers, and such that the subsequence indexed by odd (even) integers satisfies the assumptions of section 3.2 with service and inter-arrival distribution functions F 1 , G 1 (F 2 , G 2 ). Even though the moment assumptions (10)- (15) are trivially satisfied, Theorems 3 and 4 imply the existence of two different limit points for the queue length and the RES-measure in heavy traffic. In the special case when G 1 = G 2 is exponential (rate λ) distribution the above "dichotomy" is not possible, since it is well known that α(F, G) −1 = λβ/2 , where β is the second moment of F . This is an important ingredient in establishing heavy traffic behavior (16) for a sequence of M/G/1 LIFO queues, under assumptions (10)-(15). In the above example, the difference between the two α's was not large. It is plausible that α(·, ·) is a continuous function in its variables, under suitable uniform integrability assumptions. Here are some natural related open questions. How large can |α(F 1 , G 1 ) − α(F 2 , G 2 )| be under assumptions (37,38), when the mean and the variances are bounded by constants? Is it possible to construct an example of F 1 , F 2 , G 1 , G 2 , where the first k > 2 moments of F 1 and F 2 , and of G 1 and G 2 agree, but still relation (39) holds? How large can the difference |α(F 1 , G 1 ) − α(F 2 , G 2 )| be in this case? This paper concentrates on the critical case where F r = F and G r = G, for all r. It is plausible one could use a similar approach to show Conjecture 9 For a sequence of G/G/1 LIFO queues satisfying (10)- (15) and the additional assumption α(F r , G r ) → α ∈ (0, ∞) , as r → ∞ , convergence in (16) holds, where Z = α −1 W and q is defined in (17).
A person in need of a heavy traffic limit theorem for such a sequence of near critical G/G/1 queues should understand the asymptotic behavior of α(F r , G r )'s related to his/her problem. Since little is explicitly known in general about the Wiener-Hopf constants α(F, G) −1 , (unless F or G are Gamma distributions), the problem of verifying the conjecture, or its statement corresponding to a particular problem, is left to an interested reader.