scholarly journals Analysis and Computation of the Joint Queue Length Distribution in a FIFO Single-Server Queue with Multiple Batch Markovian Arrival Streams

2003 ◽  
Vol 19 (3) ◽  
pp. 349-381 ◽  
Author(s):  
Hiroyuki Masuyama ◽  
Tetsuya Takine
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Single Server ◽  
Single Server Queue ◽  
Queue Length Distribution ◽  
Server Queue

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.

On a single server queue with negative arrivals and request repeated

1999 ◽  
Vol 36 (3) ◽  
pp. 907-918 ◽  
Author(s):  
J. R. Artalejo ◽  
A. Gomez-Corral
Keyword(s):  
Queue Length ◽  
Queueing Systems ◽  
Length Distribution ◽  
Single Server ◽  
System State ◽  
Single Server Queue ◽  
Queue Length Distribution ◽  
Server Queue ◽  
Regenerative Approach ◽  
Negative Arrivals

There is a growing interest in queueing systems with negative arrivals; i.e. where the arrival of a negative customer has the effect of deleting some customer in the queue. Recently, Harrison and Pitel (1996) investigated the queue length distribution of a single server queue of type M/G/1 with negative arrivals. In this paper we extend the analysis to the context of queueing systems with request repeated. We show that the limiting distribution of the system state can still be reduced to a Fredholm integral equation. We solve such an equation numerically by introducing an auxiliary ‘truncated’ system which can easily be evaluated with the help of a regenerative approach.

On a single server queue with negative arrivals and request repeated

1999 ◽  
Vol 36 (03) ◽  
pp. 907-918 ◽  
Author(s):  
J. R. Artalejo ◽  
A. Gomez-Corral
Keyword(s):  
Queue Length ◽  
Queueing Systems ◽  
Length Distribution ◽  
Single Server ◽  
System State ◽  
Single Server Queue ◽  
Queue Length Distribution ◽  
Server Queue ◽  
Regenerative Approach ◽  
Negative Arrivals

There is a growing interest in queueing systems with negative arrivals; i.e. where the arrival of a negative customer has the effect of deleting some customer in the queue. Recently, Harrison and Pitel (1996) investigated the queue length distribution of a single server queue of type M/G/1 with negative arrivals. In this paper we extend the analysis to the context of queueing systems with request repeated. We show that the limiting distribution of the system state can still be reduced to a Fredholm integral equation. We solve such an equation numerically by introducing an auxiliary ‘truncated’ system which can easily be evaluated with the help of a regenerative approach.

scholarly journals Stationary queue length of a single-server queue with negative arrivals and nonexponential service time distributions

2018 ◽  
Vol 189 ◽  
pp. 02006 ◽  
Author(s):  
S K Koh ◽  
C H Chin ◽  
Y F Tan ◽  
L E Teoh ◽  
A H Pooi ◽  
...  
Keyword(s):  
Service Time ◽  
Queue Length ◽  
Queueing Systems ◽  
Length Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Single Server Queue ◽  
Negative Customers ◽  
Stationary Queue Length ◽  
Server Queue

In this paper, a single-server queue with negative customers is considered. The arrival of a negative customer will remove one positive customer that is being served, if any is present. An alternative approach will be introduced to derive a set of equations which will be solved to obtain the stationary queue length distribution. We assume that the service time distribution tends to a constant asymptotic rate when time t goes to infinity. This assumption will allow for finding the stationary queue length of queueing systems with non-exponential service time distributions. Numerical examples for gamma distributed service time with fractional value of shape parameter will be presented in which the steady-state distribution of queue length with such service time distributions may not be easily computed by most of the existing analytical methods.

scholarly journals Steady-state analysis of a multiserver queue in the Halfin-Whitt regime

2008 ◽  
Vol 40 (2) ◽  
pp. 548-577 ◽  
Author(s):  
David Gamarnik ◽  
Petar Momčilović
Keyword(s):  
Queue Length ◽  
Heavy Traffic ◽  
Length Distribution ◽  
Moment Generating Function ◽  
Compact Representation ◽  
Single Server ◽  
Service Time Distribution ◽  
Spare Capacity ◽  
Queue Length Distribution ◽  
Stationary Queue Length

We consider a multiserver queue in the Halfin-Whitt regime: as the number of serversngrows without a bound, the utilization approaches 1 from below at the rateAssuming that the service time distribution is lattice valued with a finite support, we characterize the limiting scaled stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain. Furthermore, we obtain an explicit expression for the critical exponent for the moment generating function of a limiting stationary queue length. This exponent has a compact representation in terms of three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime.

The steady-state solution of the M/K2/m queue

1980 ◽  
Vol 12 (03) ◽  
pp. 799-823
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Simple Formula ◽  
Length Distribution ◽  
Steady State Solution ◽  
Queue Length Distribution ◽  
Server Queue ◽  
The Mean ◽  
Mean Queue Length ◽  
The Many

The many-server queue with service time having rational Laplace transform of order 2 is considered. An expression for the asymptotic queue-length distribution is obtained. A relatively simple formula for the mean queue length is also found. A few numerical results on the mean queue length and on the probability of having to wait are given for the case of three servers. Some approximations for these quantities are also considered.

scholarly journals Joint queue length distribution of multi-class, single-server queues with preemptive priorities

2015 ◽  
Vol 81 (4) ◽  
pp. 379-395 ◽  
Author(s):  
Andrei Sleptchenko ◽  
Jori Selen ◽  
Ivo Adan ◽  
Geert-Jan van Houtum
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Single Server ◽  
Single Server Queues ◽  
Queue Length Distribution

The steady-state solution of the M/K2/m queue

1980 ◽  
Vol 12 (3) ◽  
pp. 799-823 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Simple Formula ◽  
Length Distribution ◽  
Steady State Solution ◽  
Queue Length Distribution ◽  
Server Queue ◽  
The Mean ◽  
Mean Queue Length ◽  
The Many

The many-server queue with service time having rational Laplace transform of order 2 is considered. An expression for the asymptotic queue-length distribution is obtained. A relatively simple formula for the mean queue length is also found. A few numerical results on the mean queue length and on the probability of having to wait are given for the case of three servers. Some approximations for these quantities are also considered.

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

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