scholarly journals On the Discrete-TimeGeo/G/1Queue with Vacations in Random Environment

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Jianjun Li ◽  
Liwei Liu
Keyword(s):  
Discrete Time ◽  
Random Environment ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Queue Length Distribution ◽  
Special Cases ◽  
Sojourn Time Distribution ◽  
Stationary Queue Length Distribution ◽  
Number Of Customers ◽  
Mean Time

A discrete-timeGeo/G/1queue with vacations in random environment is analyzed. Using the method of supplementary variable, we give the probability generating function (PGF) of the stationary queue length distribution at arbitrary epoch. The PGF of the stationary sojourn time distribution is also derived. And we present the various performance measures such as mean number of customers in the system, mean length of the type-icycle, and mean time that the system resides in phase0. In addition, we show that theM/G/1queue with vacations in random environment can be approximated by its discrete-time counterpart. Finally, we present some special cases of the model and numerical examples.

Traffic light queues as a generalization to queueing theory

1971 ◽  
Vol 8 (3) ◽  
pp. 480-493 ◽  
Author(s):  
Hisashi Mine ◽  
Katsuhisa Ohno
Keyword(s):  
Generating Function ◽  
Discrete Time ◽  
Queue Length ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Traffic Light ◽  
Queue Length Distribution ◽  
Mean Delay ◽  
Stationary Queue Length ◽  
Stationary Queue Length Distribution

Fixed-cycle traffic light queues have been investigated by probabilistic methods by many authors. Beckmann, McGuire and Winsten (1956) considered a discrete time queueing model with binomial arrivals and regular departure headways and derived a relation between the stationary mean delay per vehicle and the stationary mean queue-length at the beginning of a red period of the traffic light. Haight (1959) and Buckley and Wheeler (1964) considered models with Poisson arrivals and regular departure headways and investigated certain properties of the queue-length. Newell (1960) dealt with the model proposed by the first authors and obtained the probability generating function of the stationary queue-length distribution. Darroch (1964) discussed a more general discrete time model with stationary, independent arrivals and regular departure headways and derived a necessary and sufficient condition for the stationary queue-length distribution to exist and obtained its probability generating function. The above two authors, Little (1961), Miller (1963), Newell (1965), McNeil (1968), Siskind (1970) and others gave approximate expressions for the stationary mean delay per vehicle for fixed-cycle traffic light queues of various types. All of the authors mentioned above dealt with the queue-length.

An N-policy discrete-time Geo/G/1 queue with modified multiple server vacations and Bernoulli feedback*

2019 ◽  
Vol 53 (2) ◽  
pp. 367-387
Author(s):  
Shaojun Lan ◽  
Yinghui Tang
Keyword(s):  
Discrete Time ◽  
Queue Length ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Renewal Theory ◽  
Function Technique ◽  
Single Server ◽  
Bernoulli Feedback ◽  
Queue Length Distribution ◽  
Special Cases

This paper deals with a single-server discrete-time Geo/G/1 queueing model with Bernoulli feedback and N-policy where the server leaves for modified multiple vacations once the system becomes empty. Applying the law of probability decomposition, the renewal theory and the probability generating function technique, we explicitly derive the transient queue length distribution as well as the recursive expressions of the steady-state queue length distribution. Especially, some corresponding results under special cases are directly obtained. Furthermore, some numerical results are provided for illustrative purposes. Finally, a cost optimization problem is numerically analyzed under a given cost structure.

Traffic light queues as a generalization to queueing theory

1971 ◽  
Vol 8 (03) ◽  
pp. 480-493 ◽  
Author(s):  
Hisashi Mine ◽  
Katsuhisa Ohno
Keyword(s):  
Generating Function ◽  
Discrete Time ◽  
Queue Length ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Traffic Light ◽  
Queue Length Distribution ◽  
Mean Delay ◽  
Stationary Queue Length ◽  
Stationary Queue Length Distribution

Fixed-cycle traffic light queues have been investigated by probabilistic methods by many authors. Beckmann, McGuire and Winsten (1956) considered a discrete time queueing model with binomial arrivals and regular departure headways and derived a relation between the stationary mean delay per vehicle and the stationary mean queue-length at the beginning of a red period of the traffic light. Haight (1959) and Buckley and Wheeler (1964) considered models with Poisson arrivals and regular departure headways and investigated certain properties of the queue-length. Newell (1960) dealt with the model proposed by the first authors and obtained the probability generating function of the stationary queue-length distribution. Darroch (1964) discussed a more general discrete time model with stationary, independent arrivals and regular departure headways and derived a necessary and sufficient condition for the stationary queue-length distribution to exist and obtained its probability generating function. The above two authors, Little (1961), Miller (1963), Newell (1965), McNeil (1968), Siskind (1970) and others gave approximate expressions for the stationary mean delay per vehicle for fixed-cycle traffic light queues of various types. All of the authors mentioned above dealt with the queue-length.

scholarly journals The BMAP/G/1 vacation queue with queue-length dependent vacation schedule

1998 ◽  
Vol 40 (2) ◽  
pp. 207-221 ◽  
Author(s):  
Yang Woo Shin ◽  
Chareles E. M. Pearce
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Arrival Process ◽  
Service Discipline ◽  
Single Server ◽  
Server Vacation ◽  
Batch Markovian Arrival Process ◽  
Queue Length Distribution ◽  
Special Cases ◽  
Number Of Customers

AbstractWe treat a single-server vacation queue with queue-length dependent vacation schedules. This subsumes the single-server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as special cases. The lengths of vacation times depend on the number of customers in the system at the beginning of a vacation. The arrival process is a batch-Markovian arrival process (BMAP). We derive the queue-length distribution at departure epochs. By using a semi-Markov process technique, we obtain the Laplace-Stieltjes transform of the transient queue-length distribution at an arbitrary time point and its limiting distribution

scholarly journals Analysis of an N–Policy GI/M/1 Queue in a Multi–Phase Service Environmentwith Disasters

2018 ◽  
Vol 28 (2) ◽  
pp. 375-386 ◽  
Author(s):  
Tao Jiang ◽  
Sherif I. Ammar ◽  
Baoxian Chang ◽  
Liwei Liu
Keyword(s):  
Length Distribution ◽  
Threshold Value ◽  
Model Parameters ◽  
Service Environment ◽  
Queue Length Distribution ◽  
Stationary Queue Length Distribution ◽  
Service Environments ◽  
Multi Phase ◽  
The Impact ◽  
Number Of Customers

Abstract This paper investigates an N-policy GI/M/1 queue in a multi-phase service environment with disasters, where the system tends to suffer from disastrous failures while it is in operative service environments, making all present customers leave the system simultaneously and the server stop working completely. As soon as the number of customers in the queue reaches a threshold value, the server resumes its service and moves to the appropriate operative service environment immediately with some probability. We derive the stationary queue length distribution, which is then used for the computation of the Laplace-Stieltjes transform of the sojourn time of an arbitrary customer and the server’s working time in a cycle. In addition, some numerical examples are provided to illustrate the impact of several model parameters on the performance measures.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

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 A TWO-CLASS RETRIAL SYSTEM WITH COUPLED ORBIT QUEUES

2017 ◽  
Vol 31 (2) ◽  
pp. 139-179 ◽  
Author(s):  
Ioannis Dimitriou
Keyword(s):  
Performance Metrics ◽  
Kernel Method ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Single Server ◽  
Method Performance ◽  
Service Station ◽  
Queue Length Distribution

We consider a single server system accepting two types of retrial customers, which arrive according to two independent Poisson streams. The service station can handle at most one customer, and in case of blocking, typeicustomer,i=1, 2, is routed to a separate typeiorbit queue of infinite capacity. Customers from the orbits try to access the server according to the constant retrial policy. We consider coupled orbit queues, and thus, when both orbit queues are non-empty, the orbit queueitries to re-dispatch a blocked customer of typeito the main service station after an exponentially distributed time with rate μi. If an orbit queue empties, the other orbit queue changes its re-dispatch rate from μito$\mu_{i}^{\ast}$. We consider both exponential and arbitrary distributed service requirements, and show that the probability generating function of the joint stationary orbit queue length distribution can be determined using the theory of Riemann (–Hilbert) boundary value problems. For exponential service requirements, we also investigate the exact tail asymptotic behavior of the stationary joint probability distribution of the two orbits with either an idle or a busy server by using the kernel method. Performance metrics are obtained, computational issues are discussed and a simple numerical example is presented.

Transient and steady-state analysis of a queuing system having customers’ impatience with threshold

2019 ◽  
Vol 53 (5) ◽  
pp. 1861-1876 ◽  
Author(s):  
Sapana Sharma ◽  
Rakesh Kumar ◽  
Sherif Ibrahim Ammar
Keyword(s):  
Steady State ◽  
Probability Generating Function ◽  
Threshold Value ◽  
Steady State Solution ◽  
Function Technique ◽  
Single Server ◽  
Queuing Model ◽  
Queuing Models ◽  
Special Cases ◽  
Number Of Customers

In many practical queuing situations reneging and balking can only occur if the number of customers in the system is greater than a certain threshold value. Therefore, in this paper we study a single server Markovian queuing model having customers’ impatience (balking and reneging) with threshold, and retention of reneging customers. The transient analysis of the model is performed by using probability generating function technique. The expressions for the mean and variance of the number of customers in the system are obtained and a numerical example is also provided. Further the steady-state solution of the model is obtained. Finally, some important queuing models are derived as the special cases of this model.

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.

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