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.

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.

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

2008 ◽  
Vol 58 (3) ◽  
pp. 161-189 ◽  
Author(s):  
Ken’ichi Katou ◽  
Naoki Makimoto ◽  
Yukio Takahashi
Keyword(s):  
Decay Rate ◽  
Upper Bound ◽  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Queue Length Distribution

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

1979 ◽  
Vol 11 (01) ◽  
pp. 240-255 ◽  
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.

scholarly journals New Approach for Finding Basic Performance Measures of Single Server Queue

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.

scholarly journals Approximation of The Queue Length Distribution of General Queues

ETRI Journal ◽  
1994 ◽  
Vol 15 (3) ◽  
pp. 35-45 ◽  
Author(s):  
Kyu-Seok Lee ◽  
Hong Shik Park
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Queue Length Distribution

How self-similar processes persistence determines the queue length distribution moments

1999 ◽  
pp. 383-392
Author(s):  
E. Costamagna ◽  
G. Iacovoni ◽  
M. Isopi
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Queue Length Distribution ◽  
Distribution Moments ◽  
Self Similar

scholarly journals Steady-state analysis of a multiserver queue in the Halfin-Whitt regime

2008 ◽  
Vol 40 (2) ◽  
pp. 548-577 ◽  
Author(s):  
David Gamarnik ◽  
Petar Momčilović
Keyword(s):  
Queue Length ◽  
Heavy Traffic ◽  
Length Distribution ◽  
Moment Generating Function ◽  
Compact Representation ◽  
Single Server ◽  
Service Time Distribution ◽  
Spare Capacity ◽  
Queue Length Distribution ◽  
Stationary Queue Length

We consider a multiserver queue in the Halfin-Whitt regime: as the number of serversngrows without a bound, the utilization approaches 1 from below at the rateAssuming that the service time distribution is lattice valued with a finite support, we characterize the limiting scaled stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain. Furthermore, we obtain an explicit expression for the critical exponent for the moment generating function of a limiting stationary queue length. This exponent has a compact representation in terms of three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime.

Transient solutions for denumerable-state Markov processes

1994 ◽  
Vol 31 (03) ◽  
pp. 635-645
Author(s):  
Guang-Hui Hsu ◽  
Xue-Ming Yuan
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Initial Distribution ◽  
Numerical Results ◽  
Transient Solution ◽  
Queue Length Distribution ◽  
Transient Solutions

The algorithm for the transient solution for the denumerable state Markov process with an arbitrary initial distribution is given in this paper. The transient queue length distribution for a general Markovian queueing system can be obtained by this algorithm. As examples, some numerical results are presented.

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 non-exponential queueing system with independent arrivals and batch servicing

1990 ◽  
Vol 27 (02) ◽  
pp. 401-408
Author(s):  
Nico M. Van Dijk ◽  
Eric Smeitink
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Service Systems ◽  
Batch Service ◽  
Repair System ◽  
Theoretical Interest ◽  
Form Expression ◽  
Queue Length Distribution ◽  
Service Times

We study a queueing system with a finite number of input sources. Jobs are individually generated by a source but wait to be served in batches, during which the input of that source is stopped. The service speed of a server depends on the mode of other sources and thus includes interdependencies. The input and service times are allowed to be generally distributed. A classical example is a machine repair system where the machines are subject to shocks causing cumulative damage. A product-form expression is obtained for the steady state joint queue length distribution and shown to be insensitive (i.e. to depend on only mean input and service times). The result is of both practical and theoretical interest as an extension of more standard batch service systems.

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