MR/GI/1 queues by positively correlated arrival stream

1994 ◽  
Vol 31 (02) ◽  
pp. 497-514
Author(s):  
R. Szekli ◽  
R. L. Disney ◽  
S. Hur
Keyword(s):  
Renewal Process ◽  
Point Processes ◽  
Waiting Times ◽  
Jump Process ◽  
Arrival Process ◽  
Markov Renewal Process ◽  
Stochastic Comparison ◽  
Single Server ◽  
Comparison Results ◽  
The Mean

The effects of dependencies (such as association) in the arrival process to a single server queue on mean queue lengths and mean waiting times are studied. Markov renewal arrival processes with a particular transition matrix for the underlying Markov chain are used which allow us to change dependency properties without at the same time changing distributional conditions. It turns out that correlations do not seem to be pure effects, and three main factors are studied: (a) differences in the mean interarrival times in the underlying Markov renewal process, (b) intensity in the Markov renewal jump process, (c) variability in the point processes underlying the Markov renewal process. It is shown that the mean queue length can be made arbitrarily large in the class of queues with the same interarrival distributions and the same service time distributions (with fixed smaller than one traffic intensity), by making (a) large enough and (b) small enough. The existence of the moments of interest is confirmed and some stochastic comparison results for actual waiting times are shown.

MR/GI/1 queues by positively correlated arrival stream

1994 ◽  
Vol 31 (2) ◽  
pp. 497-514 ◽  
Author(s):  
R. Szekli ◽  
R. L. Disney ◽  
S. Hur
Keyword(s):  
Renewal Process ◽  
Point Processes ◽  
Waiting Times ◽  
Jump Process ◽  
Arrival Process ◽  
Markov Renewal Process ◽  
Stochastic Comparison ◽  
Single Server ◽  
Comparison Results ◽  
The Mean

The effects of dependencies (such as association) in the arrival process to a single server queue on mean queue lengths and mean waiting times are studied. Markov renewal arrival processes with a particular transition matrix for the underlying Markov chain are used which allow us to change dependency properties without at the same time changing distributional conditions. It turns out that correlations do not seem to be pure effects, and three main factors are studied: (a) differences in the mean interarrival times in the underlying Markov renewal process, (b) intensity in the Markov renewal jump process, (c) variability in the point processes underlying the Markov renewal process. It is shown that the mean queue length can be made arbitrarily large in the class of queues with the same interarrival distributions and the same service time distributions (with fixed smaller than one traffic intensity), by making (a) large enough and (b) small enough. The existence of the moments of interest is confirmed and some stochastic comparison results for actual waiting times are shown.

A new informative embedded Markov renewal process for the PH/G/1 queue

1986 ◽  
Vol 18 (02) ◽  
pp. 533-557 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Statistical Analysis ◽  
Queue Length ◽  
Busy Period ◽  
Waiting Times ◽  
Arrival Process ◽  
Embedded Markov Chain ◽  
Markov Renewal Process ◽  
The Mean ◽  
Busy Cycle

We consider a new embedded Markov chain for the PH/G/1 queue by recording the queue length, the phase of the arrival process and the number of services completed during the current busy period at the successive departure epochs. Algorithmically tractable matrix formulas are obtained which permit the analysis of the fluctuations of the queue length and waiting times during a typical busy cycle. These are useful in the computation of certain profile curves arising in the statistical analysis of queues. In addition, informative expressions for the mean waiting times in the stable GI/G/1 queue and a simple new algorithm to evaluate the waiting-time distributions for the stationary PH/PH/1 queue are obtained.

A new informative embedded Markov renewal process for the PH/G/1 queue

1986 ◽  
Vol 18 (2) ◽  
pp. 533-557 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Statistical Analysis ◽  
Queue Length ◽  
Busy Period ◽  
Waiting Times ◽  
Arrival Process ◽  
Embedded Markov Chain ◽  
Markov Renewal Process ◽  
The Mean ◽  
Busy Cycle

We consider a new embedded Markov chain for the PH/G/1 queue by recording the queue length, the phase of the arrival process and the number of services completed during the current busy period at the successive departure epochs. Algorithmically tractable matrix formulas are obtained which permit the analysis of the fluctuations of the queue length and waiting times during a typical busy cycle. These are useful in the computation of certain profile curves arising in the statistical analysis of queues. In addition, informative expressions for the mean waiting times in the stable GI/G/1 queue and a simple new algorithm to evaluate the waiting-time distributions for the stationary PH/PH/1 queue are obtained.

Stochastic comparison of repairable systems by coupling

1998 ◽  
Vol 35 (2) ◽  
pp. 348-370 ◽  
Author(s):  
Günter Last ◽  
Ryszard Szekli
Keyword(s):  
Point Processes ◽  
Stochastic Comparison ◽  
Repairable Systems ◽  
Imperfect Repair ◽  
Coupling Methods ◽  
Comparison Results ◽  
Special Cases ◽  
Repair Models

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.

scholarly journals Some performance measures for vacation models with a batch Markovian arrival process

1994 ◽  
Vol 7 (2) ◽  
pp. 111-124 ◽  
Author(s):  
Sadrac K. Matendo
Keyword(s):  
Performance Measures ◽  
Waiting Times ◽  
Distribution Functions ◽  
Markovian Arrival Process ◽  
Arrival Process ◽  
Service Discipline ◽  
Renewal Processes ◽  
Single Server ◽  
Batch Markovian Arrival Process ◽  
Interarrival Times

We consider a single server infinite capacity queueing system, where the arrival process is a batch Markovian arrival process (BMAP). Particular BMAPs are the batch Poisson arrival process, the Markovian arrival process (MAP), many batch arrival processes with correlated interarrival times and batch sizes, and superpositions of these processes. We note that the MAP includes phase-type (PH) renewal processes and non-renewal processes such as the Markov modulated Poisson process (MMPP).The server applies Kella's vacation scheme, i.e., a vacation policy where the decision of whether to take a new vacation or not, when the system is empty, depends on the number of vacations already taken in the current inactive phase. This exhaustive service discipline includes the single vacation T-policy, T(SV), and the multiple vacation T-policy, T(MV). The service times are i.i.d. random variables, independent of the interarrival times and the vacation durations. Some important performance measures such as the distribution functions and means of the virtual and the actual waiting times are given. Finally, a numerical example is presented.

A single-server queue with server vacations and a class of non-renewal arrival processes

1990 ◽  
Vol 22 (3) ◽  
pp. 676-705 ◽  
Author(s):  
David M. Lucantoni ◽  
Kathleen S. Meier-Hellstern ◽  
Marcel F. Neuts
Keyword(s):  
Waiting Times ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Arbitrary Time ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Server Queue ◽  
Arrival Processes ◽  
Markov Modulated

We study a single-server queue in which the server takes a vacation whenever the system becomes empty. The service and vacation times and the arrival process are all assumed to be mutually independent. The successive service times and the vacation times each form independent, identically distributed sequences with general distributions. A new class of non-renewal arrival processes is introduced. As special cases, it includes the Markov-modulated Poisson process and the superposition of phase-type renewal processes.Algorithmically tractable equations for the distributions of the waiting times at an arbitrary time and at arrivals, as well as for the queue length at an arbitrary time, at arrivals, and at departures are established. Some factorizations, which are known for the case of renewal input, are generalized to this new framework and new factorizations are obtained. The algorithmic implementation of these results is discussed.

Stochastic comparison of repairable systems by coupling

1998 ◽  
Vol 35 (02) ◽  
pp. 348-370 ◽  
Author(s):  
Günter Last ◽  
Ryszard Szekli
Keyword(s):  
Point Processes ◽  
Stochastic Comparison ◽  
Repairable Systems ◽  
Imperfect Repair ◽  
Coupling Methods ◽  
Comparison Results ◽  
Special Cases ◽  
Repair Models

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.

scholarly journals Pseudo-conservation laws in cyclic-service systems

1987 ◽  
Vol 24 (4) ◽  
pp. 949-964 ◽  
Author(s):  
O. J. Boxma ◽  
W. P. Groenendijk
Keyword(s):  
Conservation Laws ◽  
Conservation Law ◽  
Waiting Times ◽  
Service Systems ◽  
Stochastic Decomposition ◽  
Single Server ◽  
Weighted Sum ◽  
Cyclic Service ◽  
The Mean

This paper considers single-server, multi-queue systems with cyclic service. Non-zero switch-over times of the server between consecutive queues are assumed. A stochastic decomposition for the amount of work in such systems is obtained. This decomposition allows a short derivation of a ‘pseudo-conservation law' for a weighted sum of the mean waiting times at the various queues. Thus several recently proved conservation laws are generalised and explained.

An infinite variance solidarity theorem for Markov renewal functions

1996 ◽  
Vol 33 (2) ◽  
pp. 434-438 ◽  
Author(s):  
M. S. Sgibnev
Keyword(s):  
Renewal Process ◽  
Expected Value ◽  
Infinite Variance ◽  
Markov Renewal Process ◽  
Initial Moment ◽  
Recurrence Times ◽  
The Mean

Let , be a recurrent Markov renewal process and Mik(t) be the expected value of Nk(t) provided that at the initial moment the system is in state i. It is shown that when the mean recurrence times μ ii are finite, the differences μ ij Mki (t) – t behave asymptotically the same for all states i and k.

A generalization of little's law to moments of queue lengths and waiting times in closed, product-form queueing networks

1989 ◽  
Vol 26 (01) ◽  
pp. 121-133 ◽  
Author(s):  
James McKenna
Keyword(s):  
Sojourn Time ◽  
Queueing Networks ◽  
Waiting Times ◽  
Arrival Rate ◽  
Product Form ◽  
Single Server ◽  
Queue Lengths ◽  
The Mean ◽  
Expected Sojourn Time ◽  
Product Form Queueing Networks

Little's theorem states that under very general conditions L = λW, where L is the time average number in the system, W is the expected sojourn time in the system, and λ is the mean arrival rate to the system. For certain systems it is known that relations of the form E((L) l ) = λ lE((W) l ) are also true, where (L) l = L(L – 1)· ·· (L – l + 1). It is shown in this paper that closely analogous relations hold in closed, product-form queueing networks. Similar expressions relate Nji and Sji, where Nji is the total number of class j jobs at center i and Sji is the total sojourn time of a class j job at center i, when center i is a single-server, FCFS center. When center i is a c-server, FCFS center, Qji and Wji are related this way, where Qji is the number of class j jobs queued, but not in service at center i and Wji is the waiting time in queue of a class j job at center i. More remarkably, generalizations of these results to joint moments of queue lengths and sojourn times along overtake-free paths are shown to hold.

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