Stationary waiting-time distributions in the GI/PH/1 queue

It is known that the stable GI/PH/1 queue has an embedded Markov chain whose invariant probability vector is matrix-geometric with a rate matrix R. In terms of the matrix R, the stationary waiting-time distributions at arrivals, at an arbitrary time point and until the customer's departure may be evaluated by solving finite, highly structured systems of linear differential equations with constant coefficients. Asymptotic results, useful in truncating the computations, are also obtained. The queue discipline is first-come, first-served.

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Generalizations of the Pollaczek-Khinchin integral equation in the theory of queues

Advances in Applied Probability ◽

10.2307/1427258 ◽

1986 ◽

Vol 18 (4) ◽

pp. 952-990 ◽

Cited By ~ 36

Author(s):

Marcel F. Neuts

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Waiting Time ◽

Volterra Integral Equations ◽

Markov Renewal Process ◽

Waiting Time Distributions ◽

Virtual Waiting Time ◽

Bulk Service ◽

Constant Coefficients ◽

Linear Differential

A classical result in queueing theory states that in the stable M/G/1 queue, the stationary distribution W(x) of the waiting time of an arriving customer or of the virtual waiting time satisfies a linear Volterra integral equation of the second kind, of convolution type. For many variants of the M/G/1 queue, there are corresponding integral equations, which in most cases differ from the Pollaczek–Khinchin equation only in the form of the inhomogeneous term. This leads to interesting factorizations of the waiting-time distribution and to substantial algorithmic simplifications. In a number of priority queues, the waiting-time distributions satisfy Volterra integral equations whose kernel is a functional of the busy-period distribution in related M/G/1 queues. In other models, such as the M/G/1 queue with Bernoulli feedback or with limited admissions of customers per service, there is a more basic integral equation of Volterra type, which yields a probability distribution in terms of which the waiting-time distributions are conveniently expressed.For several complex queueing models with an embedded Markov renewal process of M/G/1 type, one obtains matrix Volterra integral equations for the waiting-time distributions or for related vectors of mass functions. Such models include the M/SM/1 and the N/G/1 queues, as well as the M/G/1 queue with some forms of bulk service.When the service-time distributions are of phase type, the numerical computation of waiting-time distributions may commonly be reduced to the solution of systems of linear differential equations with constant coefficients.

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Generalizations of the Pollaczek-Khinchin integral equation in the theory of queues

Advances in Applied Probability ◽

10.1017/s0001867800016232 ◽

1986 ◽

Vol 18 (04) ◽

pp. 952-990 ◽

Cited By ~ 9

Author(s):

Marcel F. Neuts

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Waiting Time ◽

Volterra Integral Equations ◽

Markov Renewal Process ◽

Waiting Time Distributions ◽

Virtual Waiting Time ◽

Bulk Service ◽

Constant Coefficients ◽

Linear Differential

A classical result in queueing theory states that in the stable M/G/1 queue, the stationary distribution W(x) of the waiting time of an arriving customer or of the virtual waiting time satisfies a linear Volterra integral equation of the second kind, of convolution type. For many variants of the M/G/1 queue, there are corresponding integral equations, which in most cases differ from the Pollaczek–Khinchin equation only in the form of the inhomogeneous term. This leads to interesting factorizations of the waiting-time distribution and to substantial algorithmic simplifications. In a number of priority queues, the waiting-time distributions satisfy Volterra integral equations whose kernel is a functional of the busy-period distribution in related M/G/1 queues. In other models, such as the M/G/1 queue with Bernoulli feedback or with limited admissions of customers per service, there is a more basic integral equation of Volterra type, which yields a probability distribution in terms of which the waiting-time distributions are conveniently expressed. For several complex queueing models with an embedded Markov renewal process of M/G/1 type, one obtains matrix Volterra integral equations for the waiting-time distributions or for related vectors of mass functions. Such models include the M/SM/1 and the N/G/1 queues, as well as the M/G/1 queue with some forms of bulk service. When the service-time distributions are of phase type, the numerical computation of waiting-time distributions may commonly be reduced to the solution of systems of linear differential equations with constant coefficients.

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Stationary Waiting-Time Distributions for Single-Server Queues

The Annals of Mathematical Statistics ◽

10.1214/aoms/1177704365 ◽

1962 ◽

Vol 33 (4) ◽

pp. 1323-1339 ◽

Cited By ~ 38

Author(s):

R. M. Loynes

Keyword(s):

Waiting Time ◽

Single Server ◽

Waiting Time Distributions ◽

Single Server Queues ◽

Stationary Waiting Time

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Heavy-traffic limits for polling models with exhaustive service and non-FCFS service order policies

Advances in Applied Probability ◽

10.1017/s0001867800048989 ◽

2015 ◽

Vol 47 (04) ◽

pp. 989-1014 ◽

Cited By ~ 1

Author(s):

P. Vis ◽

R. Bekker ◽

R. D. van der Mei

Keyword(s):

Waiting Time ◽

Time Distribution ◽

Heavy Traffic ◽

Processor Sharing ◽

Waiting Time Distribution ◽

Asymptotic Results ◽

Local Service ◽

Waiting Time Distributions ◽

Polling Models ◽

The Impact

We study cyclic polling models with exhaustive service at each queue under a variety of non-FCFS (first-come-first-served) local service orders, namely last-come-first-served with and without preemption, random-order-of-service, processor sharing, the multi-class priority scheduling with and without preemption, shortest-job-first, and the shortest remaining processing time policy. For each of these policies, we first express the waiting-time distributions in terms of intervisit-time distributions. Next, we use these expressions to derive the asymptotic waiting-time distributions under heavy-traffic assumptions, i.e. when the system tends to saturate. The results show that in all cases the asymptotic waiting-time distribution at queueiis fully characterized and of the form Γ Θi, with Γ and Θiindependent, and where Γ is gamma distributed with known parameters (and the same for all scheduling policies). We derive the distribution of the random variable Θiwhich explicitly expresses the impact of the local service order on the asymptotic waiting-time distribution. The results provide new fundamental insight into the impact of the local scheduling policy on the performance of a general class of polling models. The asymptotic results suggest simple closed-form approximations for the complete waiting-time distributions for stable systems with arbitrary load values.

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M/M/1 Model with Unreliable Service

International Journal of Statistics and Probability ◽

10.5539/ijsp.v7n1p125 ◽

2017 ◽

Vol 7 (1) ◽

pp. 125

Author(s):

Joshua Patterson, Andrzej Korzeniowski

Keyword(s):

Markov Chain ◽

Waiting Time ◽

Queue Length ◽

Sufficient Conditions ◽

Probability Generating Function ◽

Embedded Markov Chain ◽

Queueing Models ◽

Stieltjes Transform ◽

Stationary Queue Length ◽

Stationary Waiting Time

We define a new term ”unreliable service” and construct the corresponding embedded Markov Chain to an M/M/1 queue with so defined protocol. Sufficient conditions for positive recurrence and closed form of stationary distribution are provided. Furthermore, we compute the probability generating function of the stationary queue length and Laplace-Stieltjes transform of the stationary waiting time. In the course of the analysis an interesting decomposition of both the queue length and waiting time has emerged. A number of queueing models can be recovered from our work by taking limits of certain parameters.

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Heavy-traffic limits for polling models with exhaustive service and non-FCFS service order policies

Advances in Applied Probability ◽

10.1239/aap/1449859797 ◽

2015 ◽

Vol 47 (4) ◽

pp. 989-1014 ◽

Cited By ~ 2

Author(s):

P. Vis ◽

R. Bekker ◽

R. D. van der Mei

Keyword(s):

Waiting Time ◽

Time Distribution ◽

Heavy Traffic ◽

Processor Sharing ◽

Waiting Time Distribution ◽

Asymptotic Results ◽

Local Service ◽

Waiting Time Distributions ◽

Polling Models ◽

The Impact

We study cyclic polling models with exhaustive service at each queue under a variety of non-FCFS (first-come-first-served) local service orders, namely last-come-first-served with and without preemption, random-order-of-service, processor sharing, the multi-class priority scheduling with and without preemption, shortest-job-first, and the shortest remaining processing time policy. For each of these policies, we first express the waiting-time distributions in terms of intervisit-time distributions. Next, we use these expressions to derive the asymptotic waiting-time distributions under heavy-traffic assumptions, i.e. when the system tends to saturate. The results show that in all cases the asymptotic waiting-time distribution at queue i is fully characterized and of the form Γ Θi, with Γ and Θi independent, and where Γ is gamma distributed with known parameters (and the same for all scheduling policies). We derive the distribution of the random variable Θi which explicitly expresses the impact of the local service order on the asymptotic waiting-time distribution. The results provide new fundamental insight into the impact of the local scheduling policy on the performance of a general class of polling models. The asymptotic results suggest simple closed-form approximations for the complete waiting-time distributions for stable systems with arbitrary load values.

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REPRESENTATION OF SOLUTIONS OF SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS AND NONPERMUTABLE VARIABLE COEFFICIENTS

Mathematical Modelling and Analysis ◽

10.3846/mma.2020.11194 ◽

2020 ◽

Vol 25 (2) ◽

pp. 303-322

Author(s):

Michal Pospíšil

Keyword(s):

Differential Equations ◽

Variable Coefficients ◽

Linear Differential Equations ◽

Multiple Delays ◽

Matrix Coefficients ◽

Representation Of Solutions ◽

The Matrix ◽

Constant Coefficients ◽

Linear Differential ◽

Delay Differential

Solutions of nonhomogeneous systems of linear differential equations with multiple constant delays are explicitly stated without a commutativity assumption on the matrix coefficients. In comparison to recent results, the new formulas are not inductively built, but depend on a sum of noncommutative products in the case of constant coefficients, or on a sum of iterated integrals in the case of time-dependent coefficients. This approach shall be more suitable for applications.Representation of a solution of a Cauchy problem for a system of higher order delay differential equations is also given.

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A relation between stationary queue and waiting time distributions

Journal of Applied Probability ◽

10.1017/s0021900200035713 ◽

1971 ◽

Vol 8 (03) ◽

pp. 617-620 ◽

Cited By ~ 14

Author(s):

Rasoul Haji ◽

Gordon F. Newell

Keyword(s):

Waiting Time ◽

Queueing System ◽

Arrival Process ◽

Time Interval ◽

Queue Size ◽

Random Time Interval ◽

Queue Discipline ◽

Stationary Waiting Time ◽

Number Of Customers ◽

First In First Out

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.

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On the GI/M/1 Queue with Vacations and Multiple Service Phases

Mathematical Problems in Engineering ◽

10.1155/2017/3246934 ◽

2017 ◽

Vol 2017 ◽

pp. 1-14

Author(s):

Jianjun Li ◽

Liwei Liu

Keyword(s):

Performance Measures ◽

Waiting Time ◽

Solution Method ◽

System Size ◽

Mean Waiting Time ◽

The Matrix ◽

Matrix Geometric Solution ◽

Semi Markov Process ◽

Stationary Waiting Time ◽

Number Of Customers

This paper considers a GI/M/1 queue with vacations and multiple service phases. Whenever the system becomes empty, the server takes a vacation, causing the system to move to vacation phase 0. If the server returns from a vacation to find no customer waiting, another vacation begins. Otherwise, the system jumps from phase 0 to some service phase i with probability qi, i=1,2,…,N. Using the matrix geometric solution method and semi-Markov process, we obtain the distributions of the stationary system size at both arrival and arbitrary epochs. The distribution of the stationary waiting time of an arbitrary customer is also derived. In addition, we present some performance measures such as mean waiting time of an arbitrary customer, mean length of the type-i cycle, and mean number of customers in the system at the end of phase 0. Finally, some numerical examples are presented.

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