ON THE PROBABILITY DISTRIBUTION OF JOIN QUEUE LENGTH IN A FORK-JOIN MODEL

2010 ◽  
Vol 24 (4) ◽  
pp. 473-483
Author(s):  
Jun Li ◽  
Yiqiang Q. Zhao
Keyword(s):  
Probability Distribution ◽  
Queue Length ◽  
Length Distribution ◽  
Arrival Process ◽  
Queue Length Distribution ◽  
Poisson Arrival ◽  
Service Rates ◽  
Queue Lengths ◽  
Heterogeneous Service ◽  
Geometric Decay

In this article, we consider the two-node fork-join model with a Poisson arrival process and exponential service times of heterogeneous service rates. Using a mapping from the queue lengths in the parallel nodes to the join queue length, we first derive the probability distribution function of the join queue length in terms of joint probabilities in the parallel nodes and then study the exact tail asymptotics of the join queue length distribution. Although the asymptotics of the joint distribution of the queue lengths in the parallel nodes have three types of characterizations, our results show that the asymptotics of the join queue length distribution are characterized by two scenarios: (1) an exact geometric decay and (2) a geometric decay with the prefactor n−1/2.

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

Relationships and decomposition in the delayed bernoulli feedback queueing system

1988 ◽  
Vol 25 (1) ◽  
pp. 169-183 ◽  
Author(s):  
D. König ◽  
M. Miyazawa
Keyword(s):  
Generating Function ◽  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Arrival Process ◽  
Stieltjes Transform ◽  
Bernoulli Feedback ◽  
Queue Length Distribution ◽  
Workload Distribution ◽  
Feedback Queue

For the delayed Bernoulli feedback queue with first come–first served discipline under weak assumptions a relationship for the generating functions of the joint queue-length distribution at various points in time is given. A decomposition for the generating function of the stationary total queue length distribution has been proven. The Laplace-Stieltjes transform of the stationary joint workload distribution function is represented by its marginal distributions. The arrival process is Poisson, renewal or arbitrary stationary, respectively. The service times can form an i.i.d. sequence at each queue. Different kinds of product form of the generating function of the joint queue-length distribution are discussed.

scholarly journals On the Queue Length Distribution in BMAP Systems

2017 ◽  
Vol 2 (4) ◽  
pp. 275 ◽  
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.

On the system queue length distributions of LCFS-P queues with arbitrary acceptance and restarting policies

1992 ◽  
Vol 29 (02) ◽  
pp. 430-440 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Stationary Distribution ◽  
Queue Length ◽  
Length Distribution ◽  
Stationary System ◽  
Service Discipline ◽  
Acceptance Probability ◽  
Queue Length Distribution ◽  
Poisson Arrival ◽  
History Of

Shanthikumar and Sumita (1986) proved that the stationary system queue length distribution just after a departure instant is geometric forGI/GI/1 with LCFS-P/H service discipline and with a constant acceptance probability of an arriving customer, where P denotes preemptive and H is a restarting policy which may depend on the history of preemption. They also got interesting relationships among characteristics. We generalize those results forG/G/1 with an arbitrary restarting LCFS-P and with an arbitrary acceptance policy. Several corollaries are obtained. Fakinos' (1987) and Yamazaki's (1990) expressions for the system queue length distribution are extended. For a Poisson arrival case, we extend the well-known insensitivity for LCFS-P/resume, and discuss the stationary distribution for LCFS-P/repeat.

Relationships and decomposition in the delayed bernoulli feedback queueing system

1988 ◽  
Vol 25 (01) ◽  
pp. 169-183
Author(s):  
D. König ◽  
M. Miyazawa
Keyword(s):  
Generating Function ◽  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Arrival Process ◽  
Stieltjes Transform ◽  
Bernoulli Feedback ◽  
Queue Length Distribution ◽  
Workload Distribution ◽  
Feedback Queue

For the delayed Bernoulli feedback queue with first come–first served discipline under weak assumptions a relationship for the generating functions of the joint queue-length distribution at various points in time is given. A decomposition for the generating function of the stationary total queue length distribution has been proven. The Laplace-Stieltjes transform of the stationary joint workload distribution function is represented by its marginal distributions. The arrival process is Poisson, renewal or arbitrary stationary, respectively. The service times can form an i.i.d. sequence at each queue. Different kinds of product form of the generating function of the joint queue-length distribution are discussed.

On the system queue length distributions of LCFS-P queues with arbitrary acceptance and restarting policies

1992 ◽  
Vol 29 (2) ◽  
pp. 430-440 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Stationary Distribution ◽  
Queue Length ◽  
Length Distribution ◽  
Stationary System ◽  
Service Discipline ◽  
Acceptance Probability ◽  
Queue Length Distribution ◽  
Poisson Arrival ◽  
History Of

Shanthikumar and Sumita (1986) proved that the stationary system queue length distribution just after a departure instant is geometric for GI/GI/1 with LCFS-P/H service discipline and with a constant acceptance probability of an arriving customer, where P denotes preemptive and H is a restarting policy which may depend on the history of preemption. They also got interesting relationships among characteristics. We generalize those results for G/G/1 with an arbitrary restarting LCFS-P and with an arbitrary acceptance policy. Several corollaries are obtained. Fakinos' (1987) and Yamazaki's (1990) expressions for the system queue length distribution are extended. For a Poisson arrival case, we extend the well-known insensitivity for LCFS-P/resume, and discuss the stationary distribution for LCFS-P/repeat.

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 New Approach for Finding Basic Performance Measures of Single Server Queue

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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