Open networks of queues: their algebraic structure and estimating their transient behavior

1984 ◽  
Vol 16 (1) ◽  
pp. 176-201 ◽  
Author(s):  
William A. Massey
Keyword(s):  
Length Distribution ◽  
Transient Behavior ◽  
Product Form ◽  
General Distribution ◽  
Service Discipline ◽  
Operator Representation ◽  
Open Networks ◽  
Queue Length Distribution ◽  
Special Cases ◽  
To Receive

We develop the mathematical machinery in this paper to construct a very general class of Markovian network queueing models. Each node has a heterogeneous class of customers arriving at their own Poisson rate, ultimately to receive their own exponential service requirements. We add to this a very general type of service discipline as well as class (node) switching. These modifications allow us to model in the limit, service with a general distribution. As special cases for this model, we have the product-form networks formulated by Kelly, as well as networks with priority scheduling. For the former, we give an algebraic proof of Kelly's results for product-form networks. This is an approach that motivates the form of the solution, and justifies the various needs of local and partial balance conditions.For any network that belongs to this general model, we use the operator representation to prove stochastic dominance results. In this way, we can take the transient behavior for very complicated networks and bound its joint queue-length distribution by that for M/M/1queues.

Open networks of queues: their algebraic structure and estimating their transient behavior

1984 ◽  
Vol 16 (01) ◽  
pp. 176-201 ◽  
Author(s):  
William A. Massey
Keyword(s):  
Length Distribution ◽  
Transient Behavior ◽  
Product Form ◽  
General Distribution ◽  
Service Discipline ◽  
Operator Representation ◽  
Open Networks ◽  
Queue Length Distribution ◽  
Special Cases ◽  
To Receive

We develop the mathematical machinery in this paper to construct a very general class of Markovian network queueing models. Each node has a heterogeneous class of customers arriving at their own Poisson rate, ultimately to receive their own exponential service requirements. We add to this a very general type of service discipline as well as class (node) switching. These modifications allow us to model in the limit, service with a general distribution. As special cases for this model, we have the product-form networks formulated by Kelly, as well as networks with priority scheduling. For the former, we give an algebraic proof of Kelly's results for product-form networks. This is an approach that motivates the form of the solution, and justifies the various needs of local and partial balance conditions. For any network that belongs to this general model, we use the operator representation to prove stochastic dominance results. In this way, we can take the transient behavior for very complicated networks and bound its joint queue-length distribution by that for M/M/1queues.

scholarly journals The BMAP/G/1 vacation queue with queue-length dependent vacation schedule

1998 ◽  
Vol 40 (2) ◽  
pp. 207-221 ◽  
Author(s):  
Yang Woo Shin ◽  
Chareles E. M. Pearce
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Arrival Process ◽  
Service Discipline ◽  
Single Server ◽  
Server Vacation ◽  
Batch Markovian Arrival Process ◽  
Queue Length Distribution ◽  
Special Cases ◽  
Number Of Customers

AbstractWe treat a single-server vacation queue with queue-length dependent vacation schedules. This subsumes the single-server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as special cases. The lengths of vacation times depend on the number of customers in the system at the beginning of a vacation. The arrival process is a batch-Markovian arrival process (BMAP). We derive the queue-length distribution at departure epochs. By using a semi-Markov process technique, we obtain the Laplace-Stieltjes transform of the transient queue-length distribution at an arbitrary time point and its limiting distribution

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

2005 ◽  
Vol 42 (01) ◽  
pp. 199-222 ◽  
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.

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

1986 ◽  
Vol 23 (2) ◽  
pp. 543-549 ◽  
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.

An N-policy discrete-time Geo/G/1 queue with modified multiple server vacations and Bernoulli feedback*

2019 ◽  
Vol 53 (2) ◽  
pp. 367-387
Author(s):  
Shaojun Lan ◽  
Yinghui Tang
Keyword(s):  
Discrete Time ◽  
Queue Length ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Renewal Theory ◽  
Function Technique ◽  
Single Server ◽  
Bernoulli Feedback ◽  
Queue Length Distribution ◽  
Special Cases

This paper deals with a single-server discrete-time Geo/G/1 queueing model with Bernoulli feedback and N-policy where the server leaves for modified multiple vacations once the system becomes empty. Applying the law of probability decomposition, the renewal theory and the probability generating function technique, we explicitly derive the transient queue length distribution as well as the recursive expressions of the steady-state queue length distribution. Especially, some corresponding results under special cases are directly obtained. Furthermore, some numerical results are provided for illustrative purposes. Finally, a cost optimization problem is numerically analyzed under a given cost structure.

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

1986 ◽  
Vol 23 (02) ◽  
pp. 543-549 ◽  
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.

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

2016 ◽  
Vol 53 (4) ◽  
pp. 1125-1142 ◽  
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.

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

2005 ◽  
Vol 42 (1) ◽  
pp. 199-222 ◽  
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.

On the system queue length distributions of LCFS-P queues with arbitrary acceptance and restarting policies

1992 ◽  
Vol 29 (02) ◽  
pp. 430-440 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Stationary Distribution ◽  
Queue Length ◽  
Length Distribution ◽  
Stationary System ◽  
Service Discipline ◽  
Acceptance Probability ◽  
Queue Length Distribution ◽  
Poisson Arrival ◽  
History Of

Shanthikumar and Sumita (1986) proved that the stationary system queue length distribution just after a departure instant is geometric forGI/GI/1 with LCFS-P/H service discipline and with a constant acceptance probability of an arriving customer, where P denotes preemptive and H is a restarting policy which may depend on the history of preemption. They also got interesting relationships among characteristics. We generalize those results forG/G/1 with an arbitrary restarting LCFS-P and with an arbitrary acceptance policy. Several corollaries are obtained. Fakinos' (1987) and Yamazaki's (1990) expressions for the system queue length distribution are extended. For a Poisson arrival case, we extend the well-known insensitivity for LCFS-P/resume, and discuss the stationary distribution for LCFS-P/repeat.

scholarly journals On the Discrete-TimeGeo/G/1Queue with Vacations in Random Environment

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Jianjun Li ◽  
Liwei Liu
Keyword(s):  
Discrete Time ◽  
Random Environment ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Queue Length Distribution ◽  
Special Cases ◽  
Sojourn Time Distribution ◽  
Stationary Queue Length Distribution ◽  
Number Of Customers ◽  
Mean Time

A discrete-timeGeo/G/1queue with vacations in random environment is analyzed. Using the method of supplementary variable, we give the probability generating function (PGF) of the stationary queue length distribution at arbitrary epoch. The PGF of the stationary sojourn time distribution is also derived. And we present the various performance measures such as mean number of customers in the system, mean length of the type-icycle, and mean time that the system resides in phase0. In addition, we show that theM/G/1queue with vacations in random environment can be approximated by its discrete-time counterpart. Finally, we present some special cases of the model and numerical examples.

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