The mean waiting time of a GI/G/1 queue in light traffic via random thinning

1995 ◽  
Vol 32 (1) ◽  
pp. 256-266 ◽  
Author(s):  
Soracha Nananukul ◽  
Wei-Bo Gong
Keyword(s):  
Waiting Time ◽  
Padé Approximation ◽  
Radius Of Convergence ◽  
Arrival Process ◽  
Maclaurin Series ◽  
Light Traffic ◽  
Numerical Examples ◽  
Mean Waiting Time ◽  
The Mean ◽  
Random Thinning

In this paper, we derive the MacLaurin series of the mean waiting time in light traffic for a GI/G/1 queue. The light traffic is defined by random thinning of the arrival process. The MacLaurin series is derived with respect to the admission probability, and we prove that it has a positive radius of convergence. In the numerical examples, we use the MacLaurin series to approximate the mean waiting time beyond light traffic by means of Padé approximation.

The mean waiting time of a GI/G/1 queue in light traffic via random thinning

1995 ◽  
Vol 32 (01) ◽  
pp. 256-266
Author(s):  
Soracha Nananukul ◽  
Wei-Bo Gong
Keyword(s):  
Waiting Time ◽  
Padé Approximation ◽  
Radius Of Convergence ◽  
Arrival Process ◽  
Maclaurin Series ◽  
Light Traffic ◽  
Numerical Examples ◽  
Mean Waiting Time ◽  
The Mean ◽  
Random Thinning

In this paper, we derive the MacLaurin series of the mean waiting time in light traffic for a GI/G/1 queue. The light traffic is defined by random thinning of the arrival process. The MacLaurin series is derived with respect to the admission probability, and we prove that it has a positive radius of convergence. In the numerical examples, we use the MacLaurin series to approximate the mean waiting time beyond light traffic by means of Padé approximation.

scholarly journals Processor-sharing and random-service queues with semi-Markovian arrivals

2005 ◽  
Vol 42 (02) ◽  
pp. 478-490
Author(s):  
De-An Wu ◽  
Hideaki Takagi
Keyword(s):  
Waiting Time ◽  
Sojourn Time ◽  
Time Distribution ◽  
Arrival Process ◽  
Interarrival Time ◽  
Size Distributions ◽  
Processor Sharing ◽  
Queue Size ◽  
Mean Waiting Time ◽  
The Mean

We consider single-server queues with exponentially distributed service times, in which the arrival process is governed by a semi-Markov process (SMP). Two service disciplines, processor sharing (PS) and random service (RS), are investigated. We note that the sojourn time distribution of a type-lcustomer who, upon his arrival, meetskcustomers already present in the SMP/M/1/PS queue is identical to the waiting time distribution of a type-lcustomer who, upon his arrival, meetsk+1 customers already present in the SMP/M/1/RS queue. Two sets of system equations, one for the joint transform of the sojourn time and queue size distributions in the SMP/M/1/PS queue, and the other for the joint transform of the waiting time and queue size distributions in the SMP/M/1/RS queue, are derived. Using these equations, the mean sojourn time in the SMP/M/1/PS queue and the mean waiting time in the SMP/M/1/RS queue are obtained. We also consider a special case of the SMP in which the interarrival time distribution is determined only by the type of the customer who has most recently arrived. Numerical examples are also presented.

The mean waiting time in a G/G/m/∞ queue with the LCFS-P service discipline

1991 ◽  
Vol 23 (02) ◽  
pp. 406-428
Author(s):  
Gunter Ritter ◽  
Ulrich Wacker
Keyword(s):  
Waiting Time ◽  
Service Discipline ◽  
Mixing Condition ◽  
Input Stream ◽  
Light Traffic ◽  
Preemptive Resume ◽  
Mean Waiting Time ◽  
The Mean ◽  
Number Of Customers ◽  
Busy Cycle

A single- or multiserver queue with work-conserving service discipline and a stationary and ergodic input stream with bounded service times and arbitrarily light traffic intensity may have infinite mean waiting time. We give an example of this paradox and we also give a mixing condition which, in the case of the preemptive-resume LCFS discipline, excludes this phenomenon. Furthermore, the same methods allow to estimate the durations of the first busy period and cycle and the number of customers served in the first busy cycle of a work-conserving queue.

scholarly journals Processor-sharing and random-service queues with semi-Markovian arrivals

2005 ◽  
Vol 42 (2) ◽  
pp. 478-490 ◽  
Author(s):  
De-An Wu ◽  
Hideaki Takagi
Keyword(s):  
Waiting Time ◽  
Sojourn Time ◽  
Time Distribution ◽  
Arrival Process ◽  
Interarrival Time ◽  
Size Distributions ◽  
Processor Sharing ◽  
Queue Size ◽  
Mean Waiting Time ◽  
The Mean

We consider single-server queues with exponentially distributed service times, in which the arrival process is governed by a semi-Markov process (SMP). Two service disciplines, processor sharing (PS) and random service (RS), are investigated. We note that the sojourn time distribution of a type-l customer who, upon his arrival, meets k customers already present in the SMP/M/1/PS queue is identical to the waiting time distribution of a type-l customer who, upon his arrival, meets k+1 customers already present in the SMP/M/1/RS queue. Two sets of system equations, one for the joint transform of the sojourn time and queue size distributions in the SMP/M/1/PS queue, and the other for the joint transform of the waiting time and queue size distributions in the SMP/M/1/RS queue, are derived. Using these equations, the mean sojourn time in the SMP/M/1/PS queue and the mean waiting time in the SMP/M/1/RS queue are obtained. We also consider a special case of the SMP in which the interarrival time distribution is determined only by the type of the customer who has most recently arrived. Numerical examples are also presented.

The mean waiting time in a G/G/m/∞ queue with the LCFS-P service discipline

1991 ◽  
Vol 23 (2) ◽  
pp. 406-428 ◽  
Author(s):  
Gunter Ritter ◽  
Ulrich Wacker
Keyword(s):  
Waiting Time ◽  
Service Discipline ◽  
Mixing Condition ◽  
Input Stream ◽  
Light Traffic ◽  
Preemptive Resume ◽  
Mean Waiting Time ◽  
The Mean ◽  
Number Of Customers ◽  
Busy Cycle

A single- or multiserver queue with work-conserving service discipline and a stationary and ergodic input stream with bounded service times and arbitrarily light traffic intensity may have infinite mean waiting time. We give an example of this paradox and we also give a mixing condition which, in the case of the preemptive-resume LCFS discipline, excludes this phenomenon. Furthermore, the same methods allow to estimate the durations of the first busy period and cycle and the number of customers served in the first busy cycle of a work-conserving queue.

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

2008 ◽  
Vol 2008 ◽  
pp. 1-20 ◽  
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.

scholarly journals Estimation of the mean waiting time of a customer subject to balking: a simulation study

2007 ◽  
Vol 19 (1) ◽  
pp. 63-63
Author(s):  
Jaejin Jang ◽  
Jaewoo Chung ◽  
Jungdae Suh ◽  
Jongtae Rhee
Keyword(s):  
Waiting Time ◽  
Simulation Study ◽  
Mean Waiting Time ◽  
The Mean

scholarly journals Waiting Time Problems for Patterns in a Sequence of Multi-State Trials

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1893
Author(s):  
Bara Kim ◽  
Jeongsim Kim ◽  
Jerim Kim
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Probability Generating Function ◽  
Finite Collection ◽  
Numerical Inversion ◽  
Analytic Method ◽  
Waiting Time Distribution ◽  
Mean Waiting Time ◽  
The Matrix ◽  
The Mean

In this paper, we investigate waiting time problems for a finite collection of patterns in a sequence of independent multi-state trials. By constructing a finite GI/M/1-type Markov chain with a disaster and then using the matrix analytic method, we can obtain the probability generating function of the waiting time. From this, we can obtain the stopping probabilities and the mean waiting time, but it also enables us to compute the waiting time distribution by a numerical inversion.

scholarly journals The mean waiting time to a repetition

1983 ◽  
Vol 15 (01) ◽  
pp. 216-218
Author(s):  
Gunnar Blom
Keyword(s):  
Waiting Time ◽  
Mild Condition ◽  
Random Variables ◽  
Stationary Sequence ◽  
Mean Waiting Time ◽  
The Mean

Let X 1, X2, · ·· be a stationary sequence of random variables and E 1 , E 2 , · ··, EN mutually exclusive events defined on k consecutive X's such that the probabilities of the events have the sum unity. In the sequence E j1 , E j2 , · ·· generated by the X's, the mean waiting time from an event, say E j1 , to a repetition of that event is equal to N (under a mild condition of ergodicity). Applications are given.

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