Stability of Generalized Jackson Networks

Meyn, Sean; Down, Douglas G.

doi:10.1214/aoap/1177005203

Cited by 153 publications

(57 citation statements)

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“…These two assumptions may be relaxed; all that is necessary is that a reachable compact set exists for the Markov process (see Meyn and Down 1994 for this argument and Sigman 1990 for examples where these assumptions may be relaxed …”

Section: Stochastic Assumptionsmentioning

confidence: 99%

Compensating for Failures with Flexible Servers

Andradóttir

Ayhan

Down

2007

Operations Research

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We consider the problem of maximizing capacity in a queueing network with flexible servers, where the classes and servers are subject to failure. We assume that the interarrival and service times are independent and identically distributed, that routing is probabilistic, and that the failure state of the system can be described by a Markov process that is independent of the other system dynamics. We find that the maximal capacity is tightly bounded by the solution of a linear programming problem and that the solution of this problem can be used to construct timed, generalized round-robin policies that approach the maximal capacity arbitrarily closely. We then give a series of structural results for our policies, including identifying when server flexibility can completely compensate for failures and when the implementation of our policies can be simplified. We conclude with a numerical example that illustrates some of the developed insights.

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Section: Stochastic Assumptionsmentioning

confidence: 99%

Compensating for Failures with Flexible Servers

Andradóttir

Ayhan

Down

2007

Operations Research

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“…This is needed in Theorem 4.2 of Dai (1995), which we in turn will require in §4.1. The reader is referred to Lemma 3.4 of Meyn and Down (1994) for this argument, and to Sigman (1990) for examples where Condition (1) is not necessary. In general, we simply require that there exists a reachable compact set for the constructed Markov process, which seems reasonable, but is in practice difficult to prove.…”

Section: Stochastic Assumptionsmentioning

confidence: 99%

Dynamic Server Allocation for Queueing Networks with Flexible Servers

Andradóttir

Ayhan

Down

2003

Operations Research

142

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This paper is concerned with the design of dynamic server assignment policies that maximize the capacity of queueing networks with flexible servers. Flexibility here means that each server may be capable of performing service at several different classes in the network. We assume that the interarrival times and the service times are independent and identically distributed, and that routing is probabilistic. We also allow for server switching times, which we assume to be independent and identically distributed. We deduce the value of a tight upper bound on the achievable capacity by equating the capacity of the queueing network model with that of a limiting deterministic fluid model. The maximal capacity of the deterministic model is given by the solution to a linear programming problem that also provides optimal allocations of servers to classes. We construct particular server assignment policies, called generalized round-robin policies, that guarantee that the capacity of the queueing network will be arbitrarily close to the computed upper bound. The performance of such policies is studied using numerical examples.

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“…Dai's paper asserts that for multi-class queueing networks with an exogenous input stream the assumption (A2b), that inter-arrival times have a spread out distribution with unbounded support, implies (ii). His proof follows directly from the earlier work of Meyn and Down [24], who proved the same result for generalized Jackson networks. This needs to be extended to the case of infinite supply of work.…”

Section: B(s)mentioning

confidence: 71%

“…At the same timeQ n i (t) ⇒ 0. HenceD n I (t) converges weakly to a Brownian motion, The expression for the variance ofD I (1), (24), follows.…”

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confidence: 89%

Positive Harris recurrence and diffusion scale analysis of a push pull queueing network

Nazarathy

Weiss

2010

Performance Evaluation

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We consider a push pull queueing network with two servers and two types of jobs which are processed by the two servers in opposite order, with stochastic generally distributed processing times. This push pull network was introduced by Kopzon and Weiss, who assumed exponential processing times. It is similar to the KumarSeidman Rybko-Stolyar (KSRS) multi-class queueing network, with the distinction that instead of random arrivals, there is an infinite supply of jobs of both types. Unlike the KSRS network, we can find policies under which our push pull network works at full utilization, with both servers busy at all times, and without being congested. We perform fluid and diffusion scale analysis of this network under such policies, to show fluid stability, positive Harris recurrence, and to obtain a diffusion limit for the network. On the diffusion scale the network is empty, and the departures of the two types of jobs are highly negatively correlated Brownian motions. Using similar methods we also derive a diffusion limit of a re-entrant line with infinite supply of work.

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Stability of Generalized Jackson Networks

Cited by 153 publications

References 0 publications

Compensating for Failures with Flexible Servers

Compensating for Failures with Flexible Servers

Dynamic Server Allocation for Queueing Networks with Flexible Servers

Positive Harris recurrence and diffusion scale analysis of a push pull queueing network

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