The Annals of Applied Probability

Many-server diffusion limits for G/Ph/n+GI queues

J. G. Dai, Shuangchi He, and Tolga Tezcan

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Abstract

This paper studies many-server limits for multi-server queues that have a phase-type service time distribution and allow for customer abandonment. The first set of limit theorems is for critically loaded G/Ph/n+GI queues, where the patience times are independent and identically distributed following a general distribution. The next limit theorem is for overloaded G/Ph/n+M queues, where the patience time distribution is restricted to be exponential. We prove that a pair of diffusion-scaled total-customer-count and server-allocation processes, properly centered, converges in distribution to a continuous Markov process as the number of servers n goes to infinity. In the overloaded case, the limit is a multi-dimensional diffusion process, and in the critically loaded case, the limit is a simple transformation of a diffusion process. When the queues are critically loaded, our diffusion limit generalizes the result by Puhalskii and Reiman (2000) for GI/Ph/n queues without customer abandonment. When the queues are overloaded, the diffusion limit provides a refinement to a fluid limit and it generalizes a result by Whitt (2004) for M/M/n/+M queues with an exponential service time distribution. The proof techniques employed in this paper are innovative. First, a perturbed system is shown to be equivalent to the original system. Next, two maps are employed in both fluid and diffusion scalings. These maps allow one to prove the limit theorems by applying the standard continuous-mapping theorem and the standard random-time-change theorem.

Article information

Source
Ann. Appl. Probab. Volume 20, Number 5 (2010), 1854-1890.

Dates
First available in Project Euclid: 25 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1282747403

Digital Object Identifier
doi:10.1214/09-AAP674

Mathematical Reviews number (MathSciNet)
MR2724423

Zentralblatt MATH identifier
1202.90085

Subjects
Primary: 90B20: Traffic problems 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Keywords
Multi-server queues customer abandonment many-server heavy traffic Halfin–Whitt regime quality and efficiency-driven regime efficiency-driven regime phase-type distribution

Citation

Dai, J. G.; He, Shuangchi; Tezcan, Tolga. Many-server diffusion limits for G / Ph / n + GI queues. Ann. Appl. Probab. 20 (2010), no. 5, 1854--1890. doi:10.1214/09-AAP674. https://projecteuclid.org/euclid.aoap/1282747403


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References

  • Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • Borovkov, A. A. (1967). On limit laws for service processes in multi-channel systems. Sibirsk. Mat. J. 8 746–763.
  • Dai, J. G. and He, S. (2010). Customer abandonment in many-server queues. Math. Oper. Res. 35 347–362.
  • Dai, J. G. and Tezcan, T. (2005). State space collapse in many-server diffusion limits of parallel server systems in many-server limits. Preprint. Available at http://www2.isye.gatech.edu/~dai/publications/ssc_mor100324.pdf.
  • Dai, J. G. and Tezcan, T. (2008). Optimal control of parallel server systems with many servers in heavy traffic. Queueing Syst. 59 95–134.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Gamarnik, D. and Momčilović, P. (2008). Steady-state analysis of a multiserver queue in the Halfin–Whitt regime. Adv. in Appl. Probab. 40 548–577.
  • Garnett, O., Mandelbaum, A. and Reiman, M. (2002). Designing a call center with impatient customers. Manufacturing and Service Operations Management 4 208–227.
  • Halfin, S. and Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29 567–588.
  • Iglehart, D. L. and Whitt, W. (1970). Multiple channel queues in heavy traffic. I. Adv. in Appl. Probab. 2 150–177.
  • Jelenković, P., Mandelbaum, A. and Momčilović, P. (2004). Heavy traffic limits for queues with many deterministic servers. Queueing Syst. 47 53–69.
  • Johnson, D. P. (1983). Diffusion approximations for optimal filtering of jump processes and for queueing networks. Ph.D. thesis, Univ. Wisconsin.
  • Kang, W. N. and Ramanan, K. (2010). Fluid limits of many-server queues with reneging. Ann. Appl. Probab. To appear.
  • Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.
  • Kaspi, H. and Ramanan, K. (2010). Law of large numbers limits for many server queues. Ann. Appl. Probab. To appear.
  • Kiefer, J. and Wolfowitz, J. (1955). On the theory of queues with many servers. Trans. Amer. Math. Soc. 78 1–18.
  • Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, Philadelphia, PA.
  • Mandelbaum, A., Massey, W. A. and Reiman, M. I. (1998). Strong approximations for Markovian service networks. Queueing Syst. 30 149–201.
  • Mandelbaum, A. and Momčilović, P. (2009). Queues with many servers and impatient customers. Preprint. Available at http://iew3.technion.ac.il/serveng/References/MM0309.pdf.
  • Mandelbaum, A. and Pats, G. (1998). State-dependent stochastic networks. I. Approximations and applications with continuous diffusion limits. Ann. Appl. Probab. 8 569–646.
  • Pang, G., Talreja, R. and Whitt, W. (2007). Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probab. Surv. 4 193–267.
  • Puhalskii, A. A. and Reed, J. E. (2010). On many-server queues in heavy traffic. Ann. Appl. Probab. 20 129–195.
  • Puhalskii, A. A. and Reiman, M. I. (2000). The multiclass GI/PH/N queue in the Halfin–Whitt regime. Adv. in Appl. Probab. 32 564–595.
  • Reed, J. (2009). The G/GI/N queue in the Halfin–Whitt regime. Ann. Appl. Probab. 19 2211–2269.
  • Reiman, M. I. (1984). Open queueing networks in heavy traffic. Math. Oper. Res. 9 441–458.
  • Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Cambridge Mathematical Library 2. Cambridge Univ. Press, Cambridge.
  • Stone, C. (1963). Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7 638–660.
  • Tezcan, T. (2006). State space collapse in many server diffusion limits of parallel server systems and applications. Ph.D. thesis, School of Industrial and Systems Engineering, Georgia Institute of Technology.
  • Tezcan, T. and Dai, J. G. (2010). Dynamic control of N-systems with many servers: Asymptotic optimality of a static priority policy in heavy traffic. Oper. Res. 58 94–110.
  • Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.
  • Whitt, W. (2004). Efficiency-driven heavy-traffic approximations for many-server queues with abandonments. Manag. Sci. 50 1449–1461.
  • Whitt, W. (2005). Heavy-traffic limits for the G/H2*/n/m queue. Math. Oper. Res. 30 1–27.
  • Whitt, W. (2006). Fluid models for multiserver queues with abandonments. Oper. Res. 54 37–54.
  • Whitt, W. (2007). Proofs of the martingale FCLT. Probab. Surv. 4 268–302.
  • Zhang, J. (2009). Limited processor sharing queues and multi-server queues. Ph.D. thesis, School of Industrial and Systems Engineering, Georgia Institute of Technology.