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.

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.

New results for the Jackson network

1984 ◽  
Vol 16 (1) ◽  
pp. 7-7
Author(s):  
William A. Massey
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Decomposition Theorem ◽  
Jackson Network ◽  
Jackson Networks ◽  
Queue Length Distribution

Using operator methods, we prove a general decomposition theorem for Jackson networks. For its transient joint queue-length distribution, we can stochastically bound it above by a network that decouples into smaller independent Jackson networks.

An operator-analytic approach to the Jackson network

1984 ◽  
Vol 21 (2) ◽  
pp. 379-393 ◽  
Author(s):  
William A. Massey
Keyword(s):  
Queue Length ◽  
Heavy Traffic ◽  
Sample Path ◽  
Transient Behavior ◽  
Large Time Behavior ◽  
Total System ◽  
Time Behavior ◽  
Jackson Network ◽  
Path Ordering ◽  
Number Of Customers

Operator methods are used in this paper to systematically analyze the behavior of the Jackson network. Here, we consider rarely treated issues such as the transient behavior, and arbitrary subnetworks of the total system. By deriving the equations that govern an arbitrary subnetwork, we can see how the mean and variance for the queue length of one node as well as the covariance for two nodes vary in time.We can estimate the transient behavior by deriving a stochastic upper bound for the joint distribution of the network in terms of a judicious choice of independent M/M/1 queue-length processes. The bound we derive is one that cannot be derived by a sample-path ordering of the two processes. Moreover, we can stochastically bound from below the process for the total number of customers in the network by an M/M/1 system also. These results allow us to approximate the network by the known transient distribution of the M/M/1 queue. The bounds are tight asymptotically for large-time behavior when every node exceeds heavy-traffic conditions.

An operator-analytic approach to the Jackson network

1984 ◽  
Vol 21 (02) ◽  
pp. 379-393 ◽  
Author(s):  
William A. Massey
Keyword(s):  
Queue Length ◽  
Heavy Traffic ◽  
Sample Path ◽  
Transient Behavior ◽  
Large Time Behavior ◽  
Total System ◽  
Time Behavior ◽  
Jackson Network ◽  
Path Ordering ◽  
Number Of Customers

Operator methods are used in this paper to systematically analyze the behavior of the Jackson network. Here, we consider rarely treated issues such as the transient behavior, and arbitrary subnetworks of the total system. By deriving the equations that govern an arbitrary subnetwork, we can see how the mean and variance for the queue length of one node as well as the covariance for two nodes vary in time. We can estimate the transient behavior by deriving a stochastic upper bound for the joint distribution of the network in terms of a judicious choice of independent M/M/1 queue-length processes. The bound we derive is one that cannot be derived by a sample-path ordering of the two processes. Moreover, we can stochastically bound from below the process for the total number of customers in the network by an M/M/1 system also. These results allow us to approximate the network by the known transient distribution of the M/M/1 queue. The bounds are tight asymptotically for large-time behavior when every node exceeds heavy-traffic conditions.

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.

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.

scholarly journals Sample-path analysis of the proportional relation and its constant for discrete-time single-server queues

2006 ◽  
Vol 2006 ◽  
pp. 1-10
Author(s):  
Fumio Ishizaki ◽  
Naoto Miyoshi
Keyword(s):  
Discrete Time ◽  
Queue Length ◽  
Length Distribution ◽  
Sample Path ◽  
General Setting ◽  
Single Server ◽  
Sample Path Analysis ◽  
Stochastic Version ◽  
Queue Length Distribution ◽  
Probabilistic Setting

In the previous work, the authors have considered a discrete-time queueing system and they have established that, under some assumptions, the stationary queue length distribution for the system with capacity K1 is completely expressed in terms of the stationary distribution for the system with capacity K0 (>K1). In this paper, we study a sample-path version of this problem in more general setting, where neither stationarity nor ergodicity is assumed. We establish that, under some assumptions, the empirical queue length distribution (along through a sample path) for the system with capacity K1 is completely expressed only in terms of the quantities concerning the corresponding system with capacity K0 (>K1). Further, we consider a probabilistic setting where the assumptions are satisfied with probability one, and under the probabilistic setting, we obtain a stochastic version of our main result. The stochastic version is considered as a generalization of the author's previous result, because the probabilistic assumptions are less restrictive.

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.

scholarly journals Martingale approach for tail asymptotic problems in the gener­alized Jackson network

2018 ◽  
Vol 37 (2) ◽  
pp. 395-430 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Length Distribution ◽  
Martingale Method ◽  
Asymptotic Results ◽  
Martingale Approach ◽  
Jackson Network ◽  
Queue Length Distribution ◽  
Phase Type ◽  
Service Times ◽  
Asymptotic Problems ◽  
Tail Asymptotic

MARTINGALE APPROACH FOR TAIL ASYMPTOTIC PROBLEMS IN THE GENERALIZED JACKSON NETWORKWe study the tail asymptotic of the stationary joint queue length distribution for a generalized Jackson network GJN for short, assumingits stability. For the two-station case, this problem has recently been solved in the logarithmic sense for the marginal stationary distributions under the setting that arrival processes and service times are of phase-type. In this paper, we study similar tail asymptotic problems on the stationary distribution, but problems and assumptions are different. First, the asymptotics are studied not only for the marginal distribution but also the stationary probabilities of state sets of small volumes. Second, the interarrival and service times are generally distributed and light tailed, but of phase-type in some cases. Third, we also study the case that there are more than two stations, although the asymptotic results are less complete. For them, we develop a martingale method, which has been recently applied to a single queue with many servers by the author.

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