Networks of queues in discrete time

Abstract

Abstract:We study a new class of networks of queues whose nodes operate in round-robin fashion and other ways of interest to computer science. We compute a stationary law of product form for the Markov process describing the state of the net~vork. Moreover, we obtain the conditional expected travel time of a job given the job's requested processing times at particular nodes along its route.Zusammenfassung: Die Arbeit untersucht ein Netzwerk yon Bedienem, die nach der round-robinoder anderen Regein arbeiten, wie sie etwa bei Rechenanlagen benutzt werden. Es wird ein Markovscher Zustandsprozet~ ffir das Netzwerk definiert und dessen invariantes Gesetz angegeben. Ferner wi~d die bedingte mittlere Aufenthaltszeit eines Kunden ira Netzwerk berechnet, gegeben des Kunden Route und seine Bedienungszeitforderuv~en entlang der Route.

Discrete time queueing networks with product form steady state. Availability and performance analysis in an integrated model

“…• Discrete time analogs of Kelly networks of exponential and symmetric servers with general topology: These systems with completely reliable geometrical and doubly stochastic nodes date back to [12]; see [53] and [54] for more recent results similar to that. A survey is [10,Sect.…”

“…The geometrical servers introduced in [12] are the discrete time counterparts of Kelly's generalized exponential servers, [24,Sect. 3.1].…”

“…Each station is a geometrical server (as introduced in [12]) with an infinite waiting room under First-Come-First-Served regime (FCFS).…”

“…Remark The result of Theorem 1 is new even for the case of classical discrete time networks of completely reliable nodes; see [12,Theorem 1], [53], and [54], where the arrival streams to the network are assumed to be state independent, which simplifies analysis considerably.…”

“…We extend product form results for discrete time open queueing networks [10,12,21,43,53], to include availability of unreliable nodes and state dependent arrival intensities.…”