The synchronization of Poisson processes and queueing networks with service and synchronization nodes

This paper investigates the dynamics of a synchronization node in isolation, and of networks of service and synchronization nodes. A synchronization node consists of M infinite capacity buffers, where tokens arriving on M distinct random input flows are stored (there is one buffer for each flow). Tokens are held in the buffers until one is available from each flow. When this occurs, a token is drawn from each buffer to form a group-token, which is instantaneously released as a synchronized departure. Under independent Poisson inputs, the output of a synchronization node is shown to converge weakly (and in certain cases strongly) to a Poisson process with rate equal to the minimum rate of the input flows. Hence synchronization preserves the Poisson property, as do superposition, Bernoulli sampling and M/M/1 queueing operations. We then consider networks of synchronization and exponential server nodes with Bernoulli routeing and exogenous Poisson arrivals, extending the standard Jackson network model to include synchronization nodes. It is shown that if the synchronization skeleton of the network is acyclic (i.e. no token visits any synchronization node twice although it may visit a service node repeatedly), then the distribution of the joint queue-length process of only the service nodes is product form (under standard stability conditions) and easily computable. Moreover, the network output flows converge weakly to Poisson processes. Finally, certain results for networks with finite capacity buffers are presented, and the limiting behavior of such networks as the buffer capacities become large is studied.

[1]  Anthony Unwin,et al.  Reversibility and Stochastic Networks , 1980 .

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

[3]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[4]  W. D. Ray.  Infinitely Divisible Point Processes , 1979 .

[5]  TowsleyDon,et al.  Acyclic fork-join queuing networks , 1989 .

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

[7]  J. R. Jackson.  Networks of Waiting Lines , 1957 .

[8]  Kai Lai Chung,et al.  A Course in Probability Theory , 1949 .

[9]  J. Harrison.  Assembly-like queues , 1973, Journal of Applied Probability.

[10]  K. Mani Chandy,et al.  Open, Closed, and Mixed Networks of Queues with Different Classes of Customers , 1975, JACM.

[11]  E. CastroPeter,et al.  Infinitely Divisible Point Processes , 1982 .

[12]  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.

[13]  Gianfranco Ciardo,et al.  PNPM'91-4th International Workshop on Petri Nets and Performance Models , 1993, Performance evaluation (Print).

[14]  Venkat Anantharam,et al.  Modelling the flow of coalescing data streams through a processor , 1988 .

[15]  Armand M. Makowski,et al.  Queueing models for systems with synchronization constraints , 1989, Proc. IEEE.

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

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

[18]  Geert Jan Olsder,et al.  Synchronization and Linearity: An Algebra for Discrete Event Systems , 1994 .

[19]  P. Brémaud.  Point Processes and Queues , 1981 .

[20]  P. Brémaud.  Point processes and queues, martingale dynamics , 1983 .

[21]  Donald F. Towsley,et al.  Acyclic fork-join queuing networks , 1989, JACM.