Stabilty Conditions for Some Multiqueue Distributed Systems: Buffered Random Access Systems

Assessing stability of multidimensional systems is notoriously difficult. We consider the standard discrete-time slotted ALOHA system with a finite number of buffered users. Stability study of such a system was initiated in 1979 by Tsybakov and Mikhailov. Several bounds on the stability region have been established up-to-date, however, the exact stability region is known only for the symmetric system and two users ALOHA. This paper proves necessary and sufficient condltions for stability of the ALOHA system, hence solves the problem posed by Tsybakov and Mikhailov. We accomplish this by means of a novel technique based on three simple observations. Namely, isolating single queue from the system, applying Loynes' stability criteria for such an isolated queue, and using stochastic dominance and mathematical induction to verify the required stationarity assumptions in the Loynes' criterion. We also point out that our technique can be used to assess stability regions for other multidimensional systems. We illustrate it by providing also the stability region for a buffered system with conflict resolution algorithms. In another paper (Georgiadis and Szpankowski (1992)) we have used a similar technique to establish stability criteria for the token passing ring system. ·This research \Vas supported in part by the NSF grants CCR-8900305 and NCR-87021l5, by AFOSR grant 90-0107, and by grant ROl LM05118 from the National Library of Medicine.

[1]  A. W. Kemp,et al.  Applied Probability and Queues , 1989 .

[2]  Wojciech Szpankowski.  Bounds for Queue Lengths in a Contention Packet Broadcast System , 1986, IEEE Trans. Commun..

[3]  W. D. Ray.  Stationary Stochastic Models , 1991 .

[4]  Anthony Ephremides,et al.  On the stability of interacting queues in a multiple-access system , 1988, IEEE Trans. Inf. Theory.

[5]  W. Rosenkrantz.  Ergodicity conditions for two-dimensional Markov chains on the positive quadrant , 1989 .

[6]  R. M. Loynes,et al.  The stability of a queue with non-independent inter-arrival and service times , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  John N. Tsitsiklis,et al.  The performance of a precedence-based queuing discipline , 1986, JACM.

[8]  David J. Aldous.  Ultimate instability of exponential back-off protocol for acknowledgment-based transmission control of random access communication channels , 1987, IEEE Trans. Inf. Theory.

[9]  François Baccelli,et al.  On the stability condition of a precedence-based queueing discipline , 1989, Advances in Applied Probability.

[10]  John Capetanakis,et al.  Tree algorithms for packet broadcast channels , 1979, IEEE Trans. Inf. Theory.

[11]  A. A. Borovkov.  On the Ergodicity and Stability of the Sequence $w_{n + 1} = f(w_n ,\xi _n )$: Applications to Communication Networks , 1989 .

[12]  Panayota Papantoni-Kazakos,et al.  On the Relation Between the Finite and the Infinite Population Models for a Class of RAA's , 1987, IEEE Trans. Commun..

[13]  J. Ben Atkinson,et al.  An Introduction to Queueing Networks , 1988 .

[14]  Wojciech Szpankowski,et al.  Towards computable stability criteria for some multidimensional stochastic processes , 1989 .

[15]  P. Franken,et al.  Stationary Stochastic Models. , 1992 .

[16]  Upendra Dave,et al.  Applied Probability and Queues , 1987 .

[17]  S. Meyn,et al.  Stability of Markovian processes I: criteria for discrete-time Chains , 1992, Advances in Applied Probability.

[18]  M. Miyazawa.  The intensity conservation law for queues with randomly changed service rate , 1985, Journal of Applied Probability.

[19]  Erol Gelenbe,et al.  Stability and Optimal Control of the Packet Switching Broadcast Channel , 1977, JACM.

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

[21]  Venkat Anantharam.  The stability region of the finite-user slotted ALOHA protocol , 1991, IEEE Trans. Inf. Theory.

[22]  A. Ephremides,et al.  Ergodicity of M-dimensional random walks and random access systems , 1987, 26th IEEE Conference on Decision and Control.

[23]  R. Tweedie.  Criteria for classifying general Markov chains , 1976, Advances in Applied Probability.

[24]  Peter March,et al.  Stability of binary exponential backoff , 1988, JACM.

[25]  Frank Kelly,et al.  Stochastic Models of Computer Communication Systems , 1985 .

[26]  Guy Fayolle,et al.  On random walks arising in queueing systems: ergodicity and transience via quadratic forms as lyapounov functions — Part I , 1989, Queueing Syst. Theory Appl..

[27]  Wojciech Szpankowski,et al.  Stability of token passing rings , 1992, Queueing Syst. Theory Appl..

[28]  Wojciech Szpankowski,et al.  Stability Conditions for Multidimensional Queueing Systems with Computer Applications , 1988, Oper. Res..

[29]  Wojciech Szpankowski,et al.  On a Recurrence Equation Arising in the Analysis of Conflict Resolution Algorithms , 1987 .