On a Markovian queue with weakly correlated interarrival times

A queueing system with exponential service and correlated arrivals is analysed. Each interarrival time is exponentially distributed. The parameter of the interarrival time distribution depends on the parameter for the preceding arrival, according to a Markov chain. The parameters of the interarrival time distributions are chosen to be equal to a common value plus a factor ofε, where ε is a small number. Successive arrivals are then weakly correlated.The stability condition is found and it is shown that the system has a stationary probability vector of matrix-geometric form. Furthermore, it is shown that the stationary probabilities for the number of customers in the system, are analytic functions ofε, for sufficiently smallε, and depend more on the variability in the interarrival time distribution, than on the correlations.

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A discrete single server queue with Markovian arrivals and phase type group services

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953395000153 ◽

1995 ◽

Vol 8 (2) ◽

pp. 151-176 ◽

Cited By ~ 9

Author(s):

Attahiru Sule Alfa ◽

K. Laurie Dolhun ◽

S. Chakravarthy

Keyword(s):

Steady State ◽

Queueing System ◽

State Probability ◽

Arrival Process ◽

Single Server ◽

Probability Vector ◽

Phase Type Distribution ◽

Steady State Probability ◽

Phase Type ◽

Number Of Customers

We consider a single-server discrete queueing system in which arrivals occur according to a Markovian arrival process. Service is provided in groups of size no more than M customers. The service times are assumed to follow a discrete phase type distribution, whose representation may depend on the group size. Under a probabilistic service rule, which depends on the number of customers waiting in the queue, this system is studied as a Markov process. This type of queueing system is encountered in the operations of an automatic storage retrieval system. The steady-state probability vector is shown to be of (modified) matrix-geometric type. Efficient algorithmic procedures for the computation of the rate matrix, steady-state probability vector, and some important system performance measures are developed. The steady-state waiting time distribution is derived explicitly. Some numerical examples are presented.

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Entropy Analysis of a Flexible Markovian Queue with Server Breakdowns

Entropy ◽

10.3390/e22090979 ◽

2020 ◽

Vol 22 (9) ◽

pp. 979

Author(s):

Messaoud Bounkhel ◽

Lotfi Tadj ◽

Ramdane Hedjar

Keyword(s):

Queue Length ◽

Queueing System ◽

Threshold Level ◽

Tail Probability ◽

Probability Vector ◽

Fixed Size ◽

Markovian Queue ◽

Entropy Principle ◽

Service Rates ◽

Good Agreement

In this paper, a versatile Markovian queueing system is considered. Given a fixed threshold level c, the server serves customers one a time when the queue length is less than c, and in batches of fixed size c when the queue length is greater than or equal to c. The server is subject to failure when serving either a single or a batch of customers. Service rates, failure rates, and repair rates, depend on whether the server is serving a single customer or a batch of customers. While the analytical method provides the initial probability vector, we use the entropy principle to obtain both the initial probability vector (for comparison) and the tail probability vector. The comparison shows the results obtained analytically and approximately are in good agreement, especially when the first two moments are used in the entropy approach.

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Waiting Time in M/G/1 Queues with Impolite Arrival Disciplines

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964800003843 ◽

1995 ◽

Vol 9 (2) ◽

pp. 255-267 ◽

Cited By ~ 4

Author(s):

Süleyman Òzekici ◽

Jingwen Li ◽

Fee Seng Chou

Keyword(s):

Performance Measures ◽

Waiting Time ◽

Time Distribution ◽

Iterative Procedure ◽

Queueing System ◽

Waiting Time Distribution ◽

Special Model ◽

Average Waiting Time ◽

Random Position ◽

Number Of Customers

We consider a queueing system where arriving customers join the queue at some random position. This constitutes an impolite arrival discipline because customers do not necessarily go to the end of the line upon arrival. Although mean performance measures like the average waiting time and average number of customers in the queue are the same for all such disciplines, we show that the variance of the waiting time increases as the arrival discipline becomes more impolite, in the sense that a customer is more likely to choose a position closer to the server. For the M/G/1 model, we also provide an iterative procedure for computing the moments of the waiting time distribution. Explicit computational formulas are derived for an interesting special model where a customer joins the queue either at the head or at the end of the line.

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Queues with paired customers

Journal of Applied Probability ◽

10.2307/3213322 ◽

1981 ◽

Vol 18 (3) ◽

pp. 684-696 ◽

Cited By ~ 38

Author(s):

Guy Latouche

Keyword(s):

Queueing Systems ◽

Arrival Process ◽

Service Rate ◽

Probability Vector ◽

Simple Manner ◽

Special Service ◽

Poisson Arrivals ◽

The Difference ◽

Number Of Customers ◽

Stationary Probability Vector

Queueing systems with a special service mechanism are considered. Arrivals consist of two types of customers, and services are performed for pairs of one customer from each type. The state of the queue is described by the number of pairs and the difference, called the excess, between the number of customers of each class. Under different assumptions for the arrival process, it is shown that the excess, considered at suitably defined epochs, forms a Markov chain which is either transient or null recurrent. A system with Poisson arrivals and exponential services is then considered, for which the arrival rates depend on the excess, in such a way that the excess is bounded. It is shown that the queue is stable whenever the service rate exceeds a critical value, which depends in a simple manner on the arrival rates. For stable queues, the stationary probability vector is of matrix-geometric form and is easily computable.

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An alternative approach in finding the stationary queue length distribution of a queueing system with negative customers

ITM Web of Conferences ◽

10.1051/itmconf/20213604001 ◽

2021 ◽

Vol 36 ◽

pp. 04001

Author(s):

Siew Khew Koh ◽

Ching Herny Chin ◽

Yi Fei Tan ◽

Tan Ching Ng

Keyword(s):

Queue Length ◽

Queueing System ◽

Length Distribution ◽

Interarrival Time ◽

Negative Customers ◽

Queue Length Distribution ◽

Alternative Approach ◽

Stationary Queue Length ◽

Stationary Queue Length Distribution ◽

Stationary Probabilities

A single-server queueing system with negative customers is considered in this paper. One positive customer will be removed from the head of the queue if any negative customer is present. The distribution of the interarrival time for the positive customer is assumed to have a rate that tends to a constant as time t tends to infinity. An alternative approach will be proposed to derive a set of equations to find the stationary probabilities. The stationary probabilities will then be used to find the stationary queue length distribution. Numerical examples will be presented and compared to the results found using the analytical method and simulation procedure. The advantage of using the proposed alternative approach will be discussed in this paper.

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Queues with paired customers

Journal of Applied Probability ◽

10.1017/s0021900200098478 ◽

1981 ◽

Vol 18 (03) ◽

pp. 684-696 ◽

Cited By ~ 8

Author(s):

Guy Latouche

Keyword(s):

Queueing Systems ◽

Arrival Process ◽

Service Rate ◽

Probability Vector ◽

Simple Manner ◽

Special Service ◽

Poisson Arrivals ◽

The Difference ◽

Number Of Customers ◽

Stationary Probability Vector

Queueing systems with a special service mechanism are considered. Arrivals consist of two types of customers, and services are performed for pairs of one customer from each type. The state of the queue is described by the number of pairs and the difference, called the excess, between the number of customers of each class. Under different assumptions for the arrival process, it is shown that the excess, considered at suitably defined epochs, forms a Markov chain which is either transient or null recurrent. A system with Poisson arrivals and exponential services is then considered, for which the arrival rates depend on the excess, in such a way that the excess is bounded. It is shown that the queue is stable whenever the service rate exceeds a critical value, which depends in a simple manner on the arrival rates. For stable queues, the stationary probability vector is of matrix-geometric form and is easily computable.

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On the stability of queues with the dropping function

PLoS ONE ◽

10.1371/journal.pone.0259186 ◽

2021 ◽

Vol 16 (11) ◽

pp. e0259186

Author(s):

Andrzej Chydzinski

Keyword(s):

Queue Length ◽

Time Distribution ◽

Queueing System ◽

Discrete Event ◽

Embedded Markov Chain ◽

Service Time Distribution ◽

Poisson Arrivals ◽

The Stability ◽

Dropping Function ◽

General Service

In this paper, the stability of the queueing system with the dropping function is studied. In such system, every incoming job may be dropped randomly, with the probability being a function of the queue length. The main objective of the work is to find an easy to use condition, sufficient for the instability of the system, under assumption of Poisson arrivals and general service time distribution. Such condition is found and proven using a boundary for the dropping function and analysis of the embedded Markov chain. Applicability of the proven condition is demonstrated on several examples of dropping functions. Additionally, its correctness is confirmed using a discrete-event simulator.

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Analysis of MAP/PH(1), PH(2)/2 Queue with Bernoulli Vacations

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/2008/396871 ◽

2008 ◽

Vol 2008 ◽

pp. 1-20 ◽

Cited By ~ 3

Author(s):

B. Krishna Kumar ◽

R. Rukmani ◽

V. Thangaraj

Keyword(s):

Waiting Time ◽

Time Distribution ◽

Queueing System ◽

Arrival Process ◽

Waiting Time Distribution ◽

Steady State Probability ◽

Matrix Geometric Method ◽

Mean Waiting Time ◽

The Mean ◽

Number Of Customers

We consider a two-heterogeneous-server queueing system with Bernoulli vacation in which customers arrive according to a Markovian arrival process (MAP). Servers returning from vacation immediately take another vacation if no customer is waiting. Using matrix-geometric method, the steady-state probability of the number of customers in the system is investigated. Some important performance measures are obtained. The waiting time distribution and the mean waiting time are also discussed. Finally, some numerical illustrations are provided.

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STABILITY ANALYSIS AND SIMULATION OF N-CLASS RETRIAL SYSTEM WITH CONSTANT RETRIAL RATES AND POISSON INPUTS

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595914400028 ◽

2014 ◽

Vol 31 (02) ◽

pp. 1440002 ◽

Cited By ~ 10

Author(s):

K. AVRACHENKOV ◽

E. MOROZOV ◽

R. NEKRASOVA ◽

B. STEYAERT

Keyword(s):

Queueing System ◽

Stability Domain ◽

Single Server ◽

Retrial Queueing System ◽

Retrial Queueing ◽

Single Orbit ◽

Transmission Control Protocols ◽

Regenerative Approach ◽

The Stability ◽

Multi Access

In this paper, we study a new retrial queueing system with N classes of customers, where a class-i blocked customer joins orbit i. Orbit i works like a single-server queueing system with (exponential) constant retrial time (with rate [Formula: see text]) regardless of the orbit size. Such a system is motivated by multiple telecommunication applications, for instance wireless multi-access systems, and transmission control protocols. First, we present a review of some corresponding recent results related to a single-orbit retrial system. Then, using a regenerative approach, we deduce a set of necessary stability conditions for such a system. We will show that these conditions have a very clear probabilistic interpretation. We also performed a number of simulations to show that the obtained conditions delimit the stability domain with a remarkable accuracy, being in fact the (necessary and sufficient) stability criteria, at the very least for the 2-orbit M/M/1/1-type and M/Pareto/1/1-type retrial systems that we focus on.

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