Max-Min Optimality of Service Rate Control in Closed Queueing Networks

In this technical note, we discuss the optimality properties of service rate control in closed Jackson networks. We prove that when the cost function is linear to a particular service rate, the system performance is monotonic w.r.t. (with respect to) that service rate and the optimal value of that service rate can be either maximum or minimum (we call it Max-Min optimality); When the second-order derivative of the cost function w.r.t. a particular service rate is always positive (negative), which makes the cost function strictly convex (concave), the optimal value of such service rate for the performance maximization (minimization) problem can be either maximum or minimum. To the best of our knowledge, this is the most general result for the optimality of service rates in closed Jackson networks and all the previous works only involve the first conclusion. Moreover, our result is also valid for both the state-dependent and load-dependent service rates, under both the time-average and customer-average performance criteria.

[1]  Aurel A. Lazar,et al.  Optimal flow control of a class of queueing networks in equilibrium , 1983 .

[2]  Xi-Ren Cao,et al.  Perturbation realization, potentials, and sensitivity analysis of Markov processes , 1997, IEEE Trans. Autom. Control..

[3]  Paul Glasserman,et al.  Gradient Estimation Via Perturbation Analysis , 1990 .

[4]  Li Xia,et al.  Max-Min optimality of service rates in queueing systems with customer-average performance criterion , 2008, 2008 Winter Simulation Conference.

[5]  Xi-Ren Cao,et al.  Stochastic learning and optimization - A sensitivity-based approach , 2007, Annual Reviews in Control.

[6]  Xi-Ren Cao,et al.  Realization Probabilities: The Dynamics of Queuing Systems , 1994 .

[7]  Christos G. Cassandras,et al.  Perturbation Analysis of Discrete Event Systems , 2015, Encyclopedia of Systems and Control.

[8]  H. Scarf THE OPTIMALITY OF (S,S) POLICIES IN THE DYNAMIC INVENTORY PROBLEM , 1959 .

[9]  Rina Dechter,et al.  The optimality of A , 1988 .

[10]  Christos G. Cassandras,et al.  Introduction to Discrete Event Systems, Second Edition , 2008 .

[11]  David Yao,et al.  Decentralized control of service rates in a closed Jackson network , 1987, 26th IEEE Conference on Decision and Control.

[12]  Martin L. Puterman,et al.  Markov Decision Processes: Discrete Stochastic Dynamic Programming , 1994 .

[13]  Dye-Jyun Ma,et al.  A direct approach to decentralized control of service rates in a closed Jackson network , 1994 .

[14]  Shaler Stidham,et al.  A survey of Markov decision models for control of networks of queues , 1993, Queueing Syst. Theory Appl..

[15]  R. Weber,et al.  Optimal control of service rates in networks of queues , 1987, Advances in Applied Probability.

[16]  Christos G. Cassandras,et al.  Introduction to Discrete Event Systems , 1999, The Kluwer International Series on Discrete Event Dynamic Systems.

[17]  Xi Chen,et al.  Policy iteration for customer-average performance optimization of closed queueing systems , 2009, Autom..