New results for the Jackson network

We consider a two-node Jackson network in which the buffer of node 1 is truncated. Our interest is in the limit of the tail decay rate of the queue-length distribution of node 2 when the buffer size of node 1 goes to infinity, provided that the stability condition of the unlimited network is satisfied. We show that there can be three different cases for the limit. This generalizes some recent results obtained for the tandem Jackson network. Special cases and some numerical examples are also presented.

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A family of bounds for the transient behavior of a Jackson network

Journal of Applied Probability ◽

10.2307/3214198 ◽

1986 ◽

Vol 23 (2) ◽

pp. 543-549 ◽

Cited By ~ 4

Author(s):

William A. Massey

Keyword(s):

Length Distribution ◽

Sample Path ◽

Transient Behavior ◽

Original Network ◽

Index Set ◽

Partially Order ◽

Jackson Network ◽

Jackson Networks ◽

Queue Length Distribution ◽

Path Ordering

Using operator methods, we derive a family of stochastic bounds for the Jackson network. For its transient joint queue-length distribution, we can stochastically bound it above by various networks that decouple into smaller independent Jackson networks. Each bound is determined by a distinct partitioning of the index set for the nodes. Except for the trivial cases, none of these bounds can be extended to a sample path ordering between it and the original network. Finally, we can partially order the bounds themselves whenever one partition of the index set is the refinement of another. These results suggest new types of partial orders for stochastic processes that are not equivalent to sample-path orderings.

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A family of bounds for the transient behavior of a Jackson network

Journal of Applied Probability ◽

10.1017/s0021900200029855 ◽

1986 ◽

Vol 23 (02) ◽

pp. 543-549 ◽

Cited By ~ 4

Author(s):

William A. Massey

Keyword(s):

Length Distribution ◽

Sample Path ◽

Transient Behavior ◽

Original Network ◽

Index Set ◽

Partially Order ◽

Jackson Network ◽

Jackson Networks ◽

Queue Length Distribution ◽

Path Ordering

Using operator methods, we derive a family of stochastic bounds for the Jackson network. For its transient joint queue-length distribution, we can stochastically bound it above by various networks that decouple into smaller independent Jackson networks. Each bound is determined by a distinct partitioning of the index set for the nodes. Except for the trivial cases, none of these bounds can be extended to a sample path ordering between it and the original network. Finally, we can partially order the bounds themselves whenever one partition of the index set is the refinement of another. These results suggest new types of partial orders for stochastic processes that are not equivalent to sample-path orderings.

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Nonergodic Jackson networks with infinite supply–local stabilization and local equilibrium analysis

Journal of Applied Probability ◽

10.1017/jpr.2016.69 ◽

2016 ◽

Vol 53 (4) ◽

pp. 1125-1142 ◽

Cited By ~ 1

Author(s):

Jennifer Sommer ◽

Hans Daduna ◽

Bernd Heidergott

Keyword(s):

Performance Measures ◽

Queue Length ◽

Local Equilibrium ◽

Length Distribution ◽

Equilibrium Analysis ◽

Product Form ◽

Jackson Networks ◽

Closed Form Solutions ◽

Queue Length Distribution ◽

Infinite Supply

Abstract Classical Jackson networks are a well-established tool for the analysis of complex systems. In this paper we analyze Jackson networks with the additional features that (i) nodes may have an infinite supply of low priority work and (ii) nodes may be unstable in the sense that the queue length at these nodes grows beyond any bound. We provide the limiting distribution of the queue length distribution at stable nodes, which turns out to be of product form. A key step in establishing this result is the development of a new algorithm based on adjusted traffic equations for detecting unstable nodes. Our results complement the results known in the literature for the subcases of Jackson networks with either infinite supply nodes or unstable nodes by providing an analysis of the significantly more challenging case of networks with both types of nonstandard node present. Building on our product-form results, we provide closed-form solutions for common customer and system oriented performance measures.

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On the Effect of Finite Buffer Truncation in a Two-Node Jackson Network

Journal of Applied Probability ◽

10.1239/jap/1110381381 ◽

2005 ◽

Vol 42 (1) ◽

pp. 199-222 ◽

Cited By ~ 7

Author(s):

Yutaka Sakuma ◽

Masakiyo Miyazawa

Keyword(s):

Stability Condition ◽

Queue Length ◽

Length Distribution ◽

Buffer Size ◽

Finite Buffer ◽

Jackson Network ◽

Numerical Examples ◽

Queue Length Distribution ◽

Special Cases ◽

The Stability

We consider a two-node Jackson network in which the buffer of node 1 is truncated. Our interest is in the limit of the tail decay rate of the queue-length distribution of node 2 when the buffer size of node 1 goes to infinity, provided that the stability condition of the unlimited network is satisfied. We show that there can be three different cases for the limit. This generalizes some recent results obtained for the tandem Jackson network. Special cases and some numerical examples are also presented.

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Upper bound for the decay rate of the joint queue-length distribution in a two-node Markovian queueing system

Queueing Systems ◽

10.1007/s11134-008-9066-9 ◽

2008 ◽

Vol 58 (3) ◽

pp. 161-189 ◽

Cited By ~ 4

Author(s):

Ken’ichi Katou ◽

Naoki Makimoto ◽

Yukio Takahashi

Keyword(s):

Decay Rate ◽

Upper Bound ◽

Queue Length ◽

Queueing System ◽

Length Distribution ◽

Queue Length Distribution

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On the steady-state solution of the M/G/2 queue

Advances in Applied Probability ◽

10.1017/s0001867800031773 ◽

1979 ◽

Vol 11 (01) ◽

pp. 240-255 ◽

Cited By ~ 4

Author(s):

Per Hokstad

Keyword(s):

Steady State ◽

General Solution ◽

Laplace Transform ◽

Queue Length ◽

Joint Distribution ◽

Length Distribution ◽

Steady State Solution ◽

Queue Length Distribution ◽

The Difference ◽

Number Of Customers

The asymptotic behaviour of the M/G/2 queue is studied. The difference-differential equations for the joint distribution of the number of customers present and of the remaining holding times for services in progress were obtained in Hokstad (1978a) (for M/G/m). In the present paper it is found that the general solution of these equations involves an arbitrary function. In order to decide which of the possible solutions is the answer to the queueing problem one has to consider the singularities of the Laplace transforms involved. When the service time has a rational Laplace transform, a method of obtaining the queue length distribution is outlined. For a couple of examples the explicit form of the generating function of the queue length is obtained.

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New Approach for Finding Basic Performance Measures of Single Server Queue

Journal of Probability and Statistics ◽

10.1155/2014/851738 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11

Author(s):

Siew Khew Koh ◽

Ah Hin Pooi ◽

Yi Fei Tan

Keyword(s):

Stationary State ◽

Queue Length ◽

Length Distribution ◽

Cumulative Distribution ◽

System Capacity ◽

Interarrival Time ◽

Single Server ◽

Single Server Queue ◽

Queue Length Distribution ◽

Server Queue

Consider the single server queue in which the system capacity is infinite and the customers are served on a first come, first served basis. Suppose the probability density functionf(t)and the cumulative distribution functionF(t)of the interarrival time are such that the ratef(t)/1-F(t)tends to a constant ast→∞, and the rate computed from the distribution of the service time tends to another constant. When the queue is in a stationary state, we derive a set of equations for the probabilities of the queue length and the states of the arrival and service processes. Solving the equations, we obtain approximate results for the stationary probabilities which can be used to obtain the stationary queue length distribution and waiting time distribution of a customer who arrives when the queue is in the stationary state.

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