Time-average and asymptotically optimal flow control policies in networks with multiple transmitters

We considerM transmitting stations sending packets to a single receiver over a slotted time-multiplexed link. For each phase consisting ofT consecutive slots, the receiver dynamically allocates these slots among theM transmitters. Our objective is to characterize policies that minimize the long-term average of the total number of messages awaiting service at theM transmitters. We establish necessary and sufficient conditions on the arrival processes at the transmitters for the existence of finite cost time-average policies; it is not enough that the average arrival rate is strictly less than the slot capacity. We construct a pure strategy that attains a finite average cost under these conditions. This in turn leads to the existence of an optimal time-average pure policy for each phase lengthT, and to upper and lower bounds on the cost this policy achieves. Furthermore, we show that such an optimal time-average policy has the same properties as those of optimal discounted policies investigated by the authors in a previous paper. Finally, we prove that in the absence of costs accrued by messages within the phase, there exists a policy such that the time-average cost tends toward zero as the phase lengthT→∞.

[1]  Mario Gerla,et al.  Flow Control Protocols , 1982 .

[2]  J. Wolfowitz,et al.  On the Characteristics of the General Queueing Process, with Applications to Random Walk , 1956 .

[3]  Rodolfo A. Milito,et al.  Optimal hop-by-hop flow control policies with multiple heterogeneous transmitters , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[4]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[5]  A. Gut Stopped Random Walks , 1987 .

[6]  Zvi Rosberg,et al.  Optimal hop-by-hop flow control in computer networks , 1986 .

[7]  D. V. Lindley,et al.  The theory of queues with a single server , 1952, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  A. Gut Stopped Random Walks: Limit Theorems and Applications , 1987 .

[9]  Mario Gerla,et al.  Flow Control: A Comparative Survey , 1980, IEEE Trans. Commun..

[10]  Mischa K. Schwartz Computer Communications Network Design and Analysis , 1977 .

[11]  Pravin Varaiya,et al.  Stochastic Systems: Estimation, Identification, and Adaptive Control , 1986 .

[12]  S. Lippman On Dynamic Programming with Unbounded Rewards , 1975 .

[13]  Demosthenis Teneketzis,et al.  Properties of optimal hop-by-hop allocation policies in networks with multiple transmitters and linear equal holding costs , 1991 .

[14]  Derya Cansever,et al.  Optimal hop-by-hop control with multiple transmitters , 1987, 26th IEEE Conference on Decision and Control.

[15]  J. Kingman Inequalities in the Theory of Queues , 1970 .

[16]  J. Kiefer,et al.  On the theory of queues with many servers , 1955 .

[17]  W. Feller,et al.  An Introduction to Probability Theory and Its Application. , 1951 .

[18]  Rodolfo A. Milito,et al.  Optimal multistage hop-by-hop flow control policies: the multiple sources single destination case , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[19]  Linn I. Sennott,et al.  Average Cost Optimal Stationary Policies in Infinite State Markov Decision Processes with Unbounded Costs , 1989, Oper. Res..

[20]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .