Dynamic Scheduling of a Multiclass Queue in the Halfin-Whitt Heavy Traffic Regime

We consider a Markovian model of a multiclass queueing system in which a single large pool of servers attends to the various customer classes. Customers waiting to be served may abandon the queue, and there is a cost penalty associated with such abandonments. Service rates, abandonment rates, and abandonment penalties are generally different for the different classes. The problem studied is that of dynamically scheduling the various classes. We consider the Halfin-Whitt heavy traffic regime, where the total arrival rate and the number of servers both become large in such a way that the system's traffic intensity parameter approaches one. An approximating diffusion control problem is described and justified as a purely formal (that is, nonrigorous) heavy traffic limit. The Hamilton-Jacobi-Bellman equation associated with the limiting diffusion control problem is shown to have a smooth (classical) solution, and optimal controls are shown to have an extremal or "bang-bang" character. Several useful qualitative insights are derived from the mathematical analysis, including a "square-root rule" for sizing large systems and a sharp contrast between system behavior in the Halfin-Whitt regime versus that observed in the "conventional" heavy traffic regime. The latter phenomenon is illustrated by means of a numerical example having two customer classes.

[1]  Dimitri P. Bertsekas,et al.  Dynamic Programming and Optimal Control, Two Volume Set , 1995 .

[2]  Sunil Kumar,et al.  A numerical method for solving singular Brownian control problems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[3]  Avishai Mandelbaum,et al.  Designing a Call Center with Impatient Customers , 2002, Manuf. Serv. Oper. Manag..

[4]  M. James.  Controlled markov processes and viscosity solutions , 1994 .

[5]  Ward Whitt,et al.  Understanding the Efficiency of Multi-Server Service Systems , 1992 .

[6]  J. Michael Harrison,et al.  Dynamic Scheduling of a Multiclass Queue: Discount Optimality , 1975, Oper. Res..

[7]  Jan A. Van Mieghem,et al.  Dynamic Control of Brownian Networks: State Space Collapse and Equivalent Workload Formulations , 1997 .

[8]  J. V. Mieghem.  Dynamic Scheduling with Convex Delay Costs: The Generalized CU Rule , 1995 .

[9]  N. Krylov.  Controlled Diffusion Processes , 1980 .

[10]  J. Harrison.  Heavy traffic analysis of a system with parallel servers: asymptotic optimality of discrete-review policies , 1998 .

[11]  L. Sennott.  Stochastic Dynamic Programming and the Control of Queueing Systems , 1998 .

[12]  H. J. Kushner,et al.  Optimal Discounted Stochastic Control for Diffusion Processes , 1967 .

[13]  H. Kushner.  Numerical Methods for Stochastic Control Problems in Continuous Time , 2000 .

[14]  Sem C. Borst,et al.  Dimensioning Large Call Centers , 2000, Oper. Res..

[15]  Rami Atar,et al.  Scheduling a multi class queue with many exponential servers: asymptotic optimality in heavy traffic , 2004, math/0407058.

[16]  Avishai Mandelbaum,et al.  A Brownian control problem for a simple queueing system in the Halfin-Whitt regime , 2004, Syst. Control. Lett..

[17]  A. Bensoussan.  Stochastic control by functional analysis methods , 1982 .

[18]  W. Whitt,et al.  PIECEWISE-LINEAR DIFFUSION PROCESSES , 1995 .

[19]  W. Rudin.  Real and complex analysis , 1968 .

[20]  A. Veretennikov.  ON STRONG SOLUTIONS AND EXPLICIT FORMULAS FOR SOLUTIONS OF STOCHASTIC INTEGRAL EQUATIONS , 1981 .

[21]  J. Michael Harrison,et al.  Heavy traffic resource pooling in parallel‐server systems , 1999, Queueing Syst. Theory Appl..

[22]  O. A. Ladyzhenskai︠a︡,et al.  Linear and quasilinear elliptic equations , 1968 .

[23]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[24]  F. P. Kelly,et al.  Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling , 1993, Queueing Syst. Theory Appl..

[25]  R. J. Elliott,et al.  CONTROLLED DIFFUSION PROCESSES (Applications of Mathematics, 14) , 1981 .

[26]  J. Tinsley Oden,et al.  Finite elements: An introduction , 1991 .

[27]  R. Srikant,et al.  Diffusion approximations for a single node accessed by congestion-controlled sources , 2000, IEEE Trans. Autom. Control..

[28]  Ward Whitt,et al.  Heavy-Traffic Limits for Queues with Many Exponential Servers , 1981, Oper. Res..

[29]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[30]  A. Stolyar,et al.  Heavy tra c limits for a mobile phone system loss model , 1994 .

[31]  Philip J. Fleming,et al.  HEAVY TRAFFIC APPROXIMATIONS FOR A SYSTEM OF INFINITE SERVERS WITH LOAD BALANCING , 1999, Probability in the Engineering and Informational Sciences.

[32]  M. Reiman,et al.  The multiclass GI/PH/N queue in the Halfin-Whitt regime , 2000, Advances in Applied Probability.