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

We consider a finite-buffer single server queueing system with queue-length dependent vacations where arrivals occur according to a batch Markovian arrival process (BMAP). The service discipline is P-limited service, also called E-limited with limit variation (ELV) where the server serves until either the system is emptied or a randomly chosen limit of L customers has been served. Depending on the number of customers present in the system, the server will monitor his vacation times. Queue-length distributions at various epochs such as before, arrival, arbitrary and after, departure have been obtained. Several other service disciplines like Bernoulli scheduling, nonexhaustive service, and E-limited service can be treated as special cases of the P-limited service. Finally, the total expected cost function per unit time is considered to determine locally optimal values N* of N or a maximum limit L^* of L^ as the number of customers served during a service period at a minimum cost.

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

RAIRO - Operations Research ◽

10.1051/ro/2017027 ◽

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.

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THE QUEUE-LENGTH DISTRIBUTION FOR M x /G/1 QUEUE WITH SINGLE SERVER VACATION

Acta Mathematica Scientia ◽

10.1016/s0252-9602(17)30647-1 ◽

2000 ◽

Vol 20 (3) ◽

pp. 397-408 ◽

Cited By ~ 5

Author(s):

Yinghui Tang ◽

Xiaowo Tang

Keyword(s):

Queue Length ◽

Length Distribution ◽

Single Server ◽

Server Vacation ◽

Queue Length Distribution

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On the Queue Length Distribution in BMAP Systems

Journal of Communications Software and Systems ◽

10.24138/jcomss.v2i4.272 ◽

2017 ◽

Vol 2 (4) ◽

pp. 275 ◽

Cited By ~ 1

Author(s):

Andrzej Chydzinski

Keyword(s):

Poisson Process ◽

Queue Length ◽

Length Distribution ◽

Compound Poisson Process ◽

Arrival Process ◽

Finite Buffer ◽

Batch Markovian Arrival Process ◽

Queue Length Distribution ◽

Markovian Processes ◽

The Laplace Transform

Batch Markovian Arrival Process – BMAP – is a teletraffic model which combines high ability to imitate complexstatistical behaviour of network traces with relative simplicity in analysis and simulation. It is also a generalization of a wide class of Markovian processes, a class which in particular include the Poisson process, the compound Poisson process, the Markovmodulated Poisson process, the phase-type renewal process and others. In this paper we study the main queueing performance characteristic of a finite-buffer queue fed by the BMAP, namely the queue length distribution. In particular, we show a formula for the Laplace transform of the queue length distribution. The main benefit of this formula is that it may be used to obtain both transient and stationary characteristics. To demonstrate this, several numerical results are presented.

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Optimal control for a BMAP/G/1 queue with two service modes

Mathematical Problems in Engineering ◽

10.1155/s1024123x99001088 ◽

1999 ◽

Vol 5 (3) ◽

pp. 255-273 ◽

Cited By ~ 7

Author(s):

Alexander N. Dudin ◽

Shoichi Nishimura

Keyword(s):

Length Distribution ◽

Arrival Process ◽

Service Rate ◽

Single Server ◽

Batch Markovian Arrival Process ◽

Queue Length Distribution ◽

Telecommunication Systems ◽

Stationary Queue Length ◽

Stationary Queue Length Distribution ◽

Service Mode

Queueing models with controllable service rate play an important role in telecommunication systems. This paper deals with a single-server model with a batch Markovian arrival process (BMAP) and two service modes, where switch-over times are involved when changing the service mode. The embedded stationary queue length distribution and the explicit dependence of operation criteria on switch-over levels and derived.

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Finite capacity vacation models with non-renewal input

Journal of Applied Probability ◽

10.1017/s0021900200039516 ◽

1991 ◽

Vol 28 (01) ◽

pp. 174-197 ◽

Cited By ~ 7

Author(s):

C. Blondia

Keyword(s):

Waiting Time ◽

Time Distribution ◽

Length Distribution ◽

Time Instant ◽

Arrival Process ◽

Service Discipline ◽

Single Server ◽

Waiting Time Distribution ◽

Waiting Room ◽

Special Cases

This paper studies a single server queue with finite waiting room where the server takes vacations according to two different strategies: (i) an exhaustive service discipline, where the server takes a vacation whenever the system becomes empty and these vacations are repeated as long as there are no customers in the system upon return from a vacation, i.e. a repeated vacation strategy; (ii) a limited service discipline, where the server begins a vacation either if K customers have been served in the same busy period or if the system is empty and then a repeated vacation strategy is followed. The input process is a general Markovian arrival process introduced by Lucantoni, Meier-Hellstern and Neuts, which as special cases includes the Markov modulated Poisson process and the phase-type renewal process. The service times and vacation times each are generally distributed random variables. For both models, we obtain the queue length distribution at departures, at an arbitrary time instant and at arrival time. We also derive the loss probability of an arriving customer. We obtain formulae for the LST of the virtual waiting time distribution and for the LST of the waiting time distribution at arrival epochs.

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Finite capacity vacation models with non-renewal input

Journal of Applied Probability ◽

10.2307/3214749 ◽

1991 ◽

Vol 28 (1) ◽

pp. 174-197 ◽

Cited By ~ 23

Author(s):

C. Blondia

Keyword(s):

Waiting Time ◽

Time Distribution ◽

Length Distribution ◽

Time Instant ◽

Arrival Process ◽

Service Discipline ◽

Single Server ◽

Waiting Time Distribution ◽

Waiting Room ◽

Special Cases

This paper studies a single server queue with finite waiting room where the server takes vacations according to two different strategies: (i) an exhaustive service discipline, where the server takes a vacation whenever the system becomes empty and these vacations are repeated as long as there are no customers in the system upon return from a vacation, i.e. a repeated vacation strategy; (ii) a limited service discipline, where the server begins a vacation either if K customers have been served in the same busy period or if the system is empty and then a repeated vacation strategy is followed. The input process is a general Markovian arrival process introduced by Lucantoni, Meier-Hellstern and Neuts, which as special cases includes the Markov modulated Poisson process and the phase-type renewal process. The service times and vacation times each are generally distributed random variables. For both models, we obtain the queue length distribution at departures, at an arbitrary time instant and at arrival time. We also derive the loss probability of an arriving customer. We obtain formulae for the LST of the virtual waiting time distribution and for the LST of the waiting time distribution at arrival epochs.

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Recursive solution of queue length distribution for Geo/G/1 queue with single server vacation and variable input rate

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2011.02.018 ◽

2011 ◽

Vol 61 (9) ◽

pp. 2401-2411 ◽

Cited By ~ 3

Author(s):

Chuanyi Luo ◽

Kaili Xiang ◽

Miaomiao Yu ◽

Yinghui Tang

Keyword(s):

Queue Length ◽

Length Distribution ◽

Single Server ◽

Input Rate ◽

Server Vacation ◽

Queue Length Distribution ◽

Recursive Solution

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On the steady-state solution of the M/G/2 queue

Advances in Applied Probability ◽

10.1017/s0001867800031773 ◽

1979 ◽

Vol 11 (01) ◽

pp. 240-255 ◽

Cited By ~ 4

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.

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New Approach for Finding Basic Performance Measures of Single Server Queue

Journal of Probability and Statistics ◽

10.1155/2014/851738 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11

Author(s):

Siew Khew Koh ◽

Ah Hin Pooi ◽

Yi Fei Tan

Keyword(s):

Stationary State ◽

Queue Length ◽

Length Distribution ◽

Cumulative Distribution ◽

System Capacity ◽

Interarrival Time ◽

Single Server ◽

Single Server Queue ◽

Queue Length Distribution ◽

Server Queue

Consider the single server queue in which the system capacity is infinite and the customers are served on a first come, first served basis. Suppose the probability density functionf(t)and the cumulative distribution functionF(t)of the interarrival time are such that the ratef(t)/1-F(t)tends to a constant ast→∞, and the rate computed from the distribution of the service time tends to another constant. When the queue is in a stationary state, we derive a set of equations for the probabilities of the queue length and the states of the arrival and service processes. Solving the equations, we obtain approximate results for the stationary probabilities which can be used to obtain the stationary queue length distribution and waiting time distribution of a customer who arrives when the queue is in the stationary state.

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