Optimal Expulsion Control - A Dual Approach to Admission Control of an Ordered-Entry System

In this paper we obtain the optimal admission control policy for a first-come, first-served FCFS M/M/m ordered-entry queueing system to maximize the expected discounted and the long-run average profit reward minus cost. We introduce a new approach for determining the optimal admission control policy. The underlying idea of this approach is to construct a dual system: a preemptive last-come, first-served LCFS-P M/M/m ordered-entry queueing system that is subject to expulsion control. We show that an LCFS-P system with expulsion control is isomorphic to an FCFS system with admission control, and the two systems share the same optimal control policy. Eluding the conventional dynamic programming formulation, we approach the solution from behaviors of individual customers and their impact on the social outcome. This makes our analysis simple and intuitive and reveals a better insight into the structural properties of the optimal control policy. Besides providing formulas to compute the optimal threshold, we use the operational characteristics of the dual system to obtain an easily computable approximation for the optimal threshold. The applicability of the approach transcends well beyond the problem addressed in this paper.

[1]  I. Adler,et al.  SOCIAL OPTIMIZATION VERSUS SELF-OPTIMIZATION IN WAITING LINES , 1969 .

[2]  David D. Yao,et al.  Comparing ordered-entry queues with heterogeneous servers , 1987, Queueing Syst. Theory Appl..

[3]  Guy Pujolle,et al.  Introduction to queueing networks , 1987 .

[4]  Susan H. Xu.  A duality approach to admission and scheduling controls of queues , 1994, Queueing Syst. Theory Appl..

[5]  Emanuel Parzen,et al.  Stochastic Processes , 1962 .

[6]  Shaler Stidham,et al.  Monotonic and Insensitive Optimal Policies for Control of Queues with Undiscounted Costs , 1989, Oper. Res..

[7]  Uri Yechiali,et al.  On Optimal Balking Rules and Toll Charges in the GI/M/1 Queuing Process , 1971, Oper. Res..

[8]  J. George Shanthikumar,et al.  Optimal Dynamic Assignment of Customers to Heterogeneous Servers in Parallel , 1992, Oper. Res..

[9]  P. R. Kumar,et al.  Optimal control of a queueing system with two heterogeneous servers , 1984 .

[10]  Manfred SchÄl,et al.  Conditions for optimality in dynamic programming and for the limit of n-stage optimal policies to be optimal , 1975 .

[11]  John A. Buzacott,et al.  Stochastic models of manufacturing systems , 1993 .

[12]  Jr. Shaler Stidham.  Optimal control of admission to a queueing system , 1985 .

[13]  Susan H. Xu.  On a job resequencing issue in parallel processor stochastic scheduling , 1992 .

[14]  Shaler Stidham,et al.  Socially and Individually Optimal Control of Arrivals to a GI/M/1 Queue , 1978 .

[15]  Jean Walrand,et al.  A note on optimal control of a queuing system with two heterogeneous servers , 1984 .

[16]  Shaler Stidham,et al.  Individual versus Social Optimization in Exponential Congestion Systems , 1977, Oper. Res..

[17]  Refael Hassin,et al.  ON THE OPTIMALITY OF FIRST COME LAST SERVED QUEUES , 1985 .

[18]  S. Stidham,et al.  Individual versus Social Optimization in the Allocation of Customers to Alternative Servers , 1983 .

[19]  David D. Yao.  The Arrangement of Servers in an Ordered-Entry System , 1987, Oper. Res..

[20]  B. Doshi.  Continuous time control of the arrival process in an m/g/1 queue , 1977 .

[21]  U. Yechiali.  Customers' Optimal Joining Rules for the GI/M/s Queue , 1972 .

[22]  S. Stidham,et al.  Control of arrivals to a stochastic input–output system , 1980, Advances in Applied Probability.

[23]  Niels Chr. Knudsen.  Individual and Social Optimization in a Multiserver Queue with a General Cost-Benefit Structure , 1972 .

[24]  P. Naor.  The Regulation of Queue Size by Levying Tolls , 1969 .

[25]  K. Waldmann,et al.  Optimal control of arrivals to multiserver queues in a random environment , 1984 .

[26]  Steven A. Lippman,et al.  Applying a New Device in the Optimization of Exponential Queuing Systems , 1975, Oper. Res..

[27]  Ronald W. Wolff,et al.  Stochastic Modeling and the Theory of Queues , 1989 .