STATIONARY WAITING TIME IN A QUEUEING SYSTEM WITH INVERSE SERVICE ORDER AND GENERALIZED PROBABILISTIC PRIORITY

2015 ◽  
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Stationary Waiting Time

A relation between stationary queue and waiting time distributions

1971 ◽  
Vol 8 (03) ◽  
pp. 617-620 ◽  
Author(s):  
Rasoul Haji ◽  
Gordon F. Newell
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Arrival Process ◽  
Time Interval ◽  
Queue Size ◽  
Random Time Interval ◽  
Queue Discipline ◽  
Stationary Waiting Time ◽  
Number Of Customers ◽  
First In First Out

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.

A relation between stationary queue and waiting time distributions

1971 ◽  
Vol 8 (3) ◽  
pp. 617-620 ◽  
Author(s):  
Rasoul Haji ◽  
Gordon F. Newell
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Arrival Process ◽  
Time Interval ◽  
Queue Size ◽  
Random Time Interval ◽  
Queue Discipline ◽  
Stationary Waiting Time ◽  
Number Of Customers ◽  
First In First Out

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.

scholarly journals Decomposition of M/M/1 With Unreliable Service and a Working Vacation

2020 ◽  
Vol 9 (1) ◽  
pp. 63
Author(s):  
Joshua Patterson ◽  
Andrzej Korzeniowski
Keyword(s):  
Closed Form ◽  
Failure Rate ◽  
Waiting Time ◽  
Queue Length ◽  
Service Failure ◽  
Working Vacation ◽  
Stationary Queue Length ◽  
Stationary Waiting Time ◽  
Relationship Of ◽  
The Relationship

We use the stationary distribution for the M/M/1 with Unreliable Service and aWorking Vacation (M/M/1/US/WV) given explicitly in (Patterson & Korzeniowski, 2019) to find a decomposition of the stationary queue length N. By applying the distributional form of Little's Law the Laplace-tieltjes Transform of the stationary customer waiting time W is derived. The closed form of the expected value and variance for both N and W is found and the relationship of the expected stationary waiting time as a function of the service failure rate is determined.

scholarly journals Single server queues with modified service mechanisms

1962 ◽  
Vol 2 (4) ◽  
pp. 499-507 ◽  
Author(s):  
G. F. Yeo
Keyword(s):  
Time Distribution ◽  
Busy Period ◽  
Queueing System ◽  
Probability Generating Function ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Period Distribution ◽  
Time Dependent Problem ◽  
Stationary Waiting Time ◽  
Special Case

SummaryThis paper considers a generalisation of the queueing system M/G/I, where customers arriving at empty and non-empty queues have different service time distributions. The characteristic function (c.f.) of the stationary waiting time distribution and the probability generating function (p.g.f.) of the queue size are obtained. The busy period distribution is found; the results are generalised to an Erlangian inter-arrival distribution; the time-dependent problem is considered, and finally a special case of server absenteeism is discussed.

On the waiting time distribution in a generalized queueing system with uniformly bounded sojourn times

1972 ◽  
Vol 9 (3) ◽  
pp. 642-649 ◽  
Author(s):  
Jacqueline Loris-Teghem
Keyword(s):  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Queueing System ◽  
Waiting Time Distribution ◽  
Stieltjes Transform ◽  
Sojourn Times ◽  
Stieltjes Transforms ◽  
Uniformly Bounded

A generalized queueing system with (N + 2) types of triplets (delay, service time, probability of joining the queue) and with uniformly bounded sojourn times is considered. An expression for the generating function of the Laplace-Stieltjes transforms of the waiting time distributions is derived analytically, in a case where some of the random variables defining the model have a rational Laplace-Stieltjes transform.The standard Kl/Km/1 queueing system with uniformly bounded sojourn times is considered in particular.

Another martingale bound on the waiting-time distribution in GI/G/1 queues

1979 ◽  
Vol 16 (2) ◽  
pp. 454-457 ◽  
Author(s):  
Harry H. Tan
Keyword(s):  
Waiting Time ◽  
Upper Bound ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Martingale Approach ◽  
Stationary Waiting Time

A new upper bound on the stationary waiting-time distribution of a GI/G/1 queue is derived following Kingman's martingale approach. This bound is generally stronger than Kingman's upper bound and is sometimes stronger than an upper bound derived by Ross.

On the comparison of waiting times in tandem queues

1981 ◽  
Vol 18 (03) ◽  
pp. 707-714 ◽  
Author(s):  
Shun-Chen Niu
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Waiting Times ◽  
Distribution Functions ◽  
Partial Ordering ◽  
Tandem Queues ◽  
Order Relations ◽  
Service Distribution ◽  
In Series ◽  
Definition Of

Using a definition of partial ordering of distribution functions, it is proven that for a tandem queueing system with many stations in series, where each station can have either one server with an arbitrary service distribution or a number of constant servers in parallel, the expected total waiting time in system of every customer decreases as the interarrival and service distributions becomes smaller with respect to that ordering. Some stronger conclusions are also given under stronger order relations. Using these results, bounds for the expected total waiting time in system are then readily obtained for wide classes of tandem queues.

On a property of a refusals stream

1997 ◽  
Vol 34 (03) ◽  
pp. 800-805 ◽  
Author(s):  
Vyacheslav M. Abramov
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Busy Period ◽  
Elementary Proof ◽  
Queueing System ◽  
Queueing Systems ◽  
Single Server ◽  
Wide Family

This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.

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