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.

scholarly journals Pseudo-conservation laws in cyclic-service systems

1987 ◽  
Vol 24 (04) ◽  
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.

A conservation law for single-server queues and its applications

1991 ◽  
Vol 28 (1) ◽  
pp. 198-209 ◽  
Author(s):  
Genji Yamazaki ◽  
Hirotaka Sakasegawa ◽  
J. George Shanthikumar
Keyword(s):  
Conservation Law ◽  
Extremal Property ◽  
Service Discipline ◽  
Single Server ◽  
Processor Sharing ◽  
Equilibrium Equations ◽  
Common Mean ◽  
The Mean ◽  
Mean Queue Length ◽  
Basic Equations

We establish a conservation law for G/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in a G/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in a GI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

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.

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.

scholarly journals A Unifying Conservation Law for Single-Server Queues

2007 ◽  
Vol 44 (04) ◽  
pp. 1078-1087 ◽  
Author(s):  
Urtzi Ayesta
Keyword(s):  
Conservation Laws ◽  
Service Time ◽  
Conservation Law ◽  
Single Server ◽  
Processor Sharing ◽  
Time Sharing ◽  
Single Class ◽  
Class Time ◽  
Single Server Queues ◽  
General Service

We develop a conservation law for a multi-class GI/GI/1 queue operating under a general work-conserving scheduling discipline. For single-class single-server queues, conservation laws have been obtained for both nonanticipating and anticipating disciplines with general service time distributions. For multi-class single-server queues, conservation laws have been obtained for (i) nonanticipating disciplines with exponential service time distributions and (ii) nonpreemptive nonanticipating disciplines with general service time distributions. The unifying conservation law we develop generalizes already existing conservation laws. In addition, it covers popular nonanticipating multi-class time-sharing disciplines such as discriminatory processor sharing (DPS) and generalized processor sharing (GPS) with general service time distributions. As an application, we show that the unifying conservation law can be used to compare the expected unconditional response time under two scheduling disciplines.

scholarly journals Waiting times in discrete-time cyclic-service systems

1988 ◽  
Vol 36 (2) ◽  
pp. 164-170 ◽  
Author(s):  
O.J. Boxma ◽  
W.P. Groenendijk
Keyword(s):  
Discrete Time ◽  
Waiting Times ◽  
Service Systems ◽  
Cyclic Service

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

1989 ◽  
Vol 26 (1) ◽  
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.

A conservation law for single-server queues and its applications

1991 ◽  
Vol 28 (01) ◽  
pp. 198-209 ◽  
Author(s):  
Genji Yamazaki ◽  
Hirotaka Sakasegawa ◽  
J. George Shanthikumar
Keyword(s):  
Conservation Law ◽  
Extremal Property ◽  
Service Discipline ◽  
Single Server ◽  
Processor Sharing ◽  
Equilibrium Equations ◽  
Common Mean ◽  
The Mean ◽  
Mean Queue Length ◽  
Basic Equations

We establish a conservation law forG/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in aG/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in aGI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

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.

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