The G/M/m queue with finite waiting room

1975 ◽  
Vol 12 (4) ◽  
pp. 779-792 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Asymptotic Distribution ◽  
Waiting Time ◽  
Joint Distribution ◽  
Transient Solution ◽  
Waiting Room ◽  
Waiting Time Distributions ◽  
Virtual Waiting Time ◽  
The Moment ◽  
Number Of Customers

The G/M/m queue with only s waiting places is studied. We start by studying the joint distribution of the number of customers present at time t and the time elapsing until the next arrival after t. This gives the asymptotic distribution of the number of customers at the moment of an arrival and at an arbitrary moment. Then waiting time and virtual waiting time distributions are easily obtained. For the G/M/1 queue also the transient solution is given. Finally the case s = ∞ is considered.

The G/M/m queue with finite waiting room

1975 ◽  
Vol 12 (04) ◽  
pp. 779-792 ◽  
Author(s):  
Per Hokstad
Keyword(s):  
Asymptotic Distribution ◽  
Waiting Time ◽  
Joint Distribution ◽  
Transient Solution ◽  
Waiting Room ◽  
Waiting Time Distributions ◽  
Virtual Waiting Time ◽  
The Moment ◽  
Number Of Customers

The G/M/m queue with only s waiting places is studied. We start by studying the joint distribution of the number of customers present at time t and the time elapsing until the next arrival after t. This gives the asymptotic distribution of the number of customers at the moment of an arrival and at an arbitrary moment. Then waiting time and virtual waiting time distributions are easily obtained. For the G/M/1 queue also the transient solution is given. Finally the case s = ∞ is considered.

Some general results for many server queues

1973 ◽  
Vol 5 (01) ◽  
pp. 153-169 ◽  
Author(s):  
J. H. A. De Smit
Keyword(s):  
Integral Equations ◽  
Waiting Time ◽  
Queue Length ◽  
Joint Distribution ◽  
Waiting Times ◽  
Virtual Waiting Time ◽  
Server Queue ◽  
Many Server Queues ◽  
The Many ◽  
Busy Cycle

Pollaczek's theory for the many server queue is generalized and extended. Pollaczek (1961) found the distribution of the actual waiting times in the model G/G/s as a solution of a set of integral equations. We give a somewhat more general set of integral equations from which the joint distribution of the actual waiting time and some other random variables may be found. With this joint distribution we can obtain distributions of a number of characteristic quantities, such as the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. For a wide class of many server queues the formal expressions may lead to explicit results.

scholarly journals The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1

2002 ◽  
Vol 39 (03) ◽  
pp. 619-629 ◽  
Author(s):  
Gang Uk Hwang ◽  
Bong Dae Choi ◽  
Jae-Kyoon Kim
Keyword(s):  
Discrete Time ◽  
Waiting Time ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Autoregressive Process ◽  
Waiting Time Distribution ◽  
Tail Probabilities ◽  
Waiting Time Distributions ◽  
Virtual Waiting Time ◽  
Asymptotic Decay Rate

We consider a discrete-time queueing system with the discrete autoregressive process of order 1 (DAR(1)) as an input process and obtain the actual waiting time distribution and the virtual waiting time distribution. As shown in the analysis, our approach provides a natural numerical algorithm to compute the waiting time distributions, based on the theory of the GI/G/1 queue, and consequently we can easily investigate the effect of the parameters of the DAR(1) on the waiting time distributions. We also derive a simple approximation of the asymptotic decay rate of the tail probabilities for the virtual waiting time in the heavy traffic case.

The cyclic queue with one general and one exponential server

1983 ◽  
Vol 15 (04) ◽  
pp. 857-873 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Time Distribution ◽  
Joint Distribution ◽  
Present Model ◽  
Response Times ◽  
Strong Connection ◽  
Waiting Room ◽  
Successive Response ◽  
Proportional Rate ◽  
The Mean ◽  
Number Of Customers

This paper considers the two-stage cyclic queueing model consisting of one general (G) and one exponential (M) server. The strong connection between the present model and the M/G/1 model (with finite waiting room) is exploited to yield the joint distribution of the successive response times of a customer at the G queue and the M queue. This result reveals a surprising phenomenon: in general there is a difference between the joint distribution of the two successive response times at (first) the G queue and (then) the M queue, and the joint distribution of the two successive response times at (first) the M queue and (then) the G queue. Another associated result is an expression for the cycle-time distribution. Special consideration is given to the case that the number of customers in the system tends to ∞, while the mean service times tend to 0 at an inversely proportional rate.

Comparing multi-server queues with finite waiting rooms, II: Different numbers of servers

1979 ◽  
Vol 11 (02) ◽  
pp. 448-455 ◽  
Author(s):  
David Sonderman
Keyword(s):  
Probability Space ◽  
Queueing Systems ◽  
Stochastic Order ◽  
Sample Path ◽  
System Size ◽  
Waiting Room ◽  
Virtual Waiting Time ◽  
Number Of Customers ◽  
Multi Server ◽  
Quantities Of Interest

We compare two queueing systems with identical general arrival streams, but different numbers of servers, different waiting room capacities, and stochastically ordered service time distributions. Under appropriate conditions, it is possible to construct two new systems on the same probability space so that the new systems are probabilistically equivalent to the original systems and each sample path of the stochastic process representing system size in one system lies entirely below the corresponding sample path in the other system. This construction implies stochastic order for these processes and many associated quantities of interest, such as a busy period, the number of customers lost in any interval, and the virtual waiting time.

Some general results for many server queues

1973 ◽  
Vol 5 (1) ◽  
pp. 153-169 ◽  
Author(s):  
J. H. A. De Smit
Keyword(s):  
Integral Equations ◽  
Waiting Time ◽  
Queue Length ◽  
Joint Distribution ◽  
Waiting Times ◽  
Virtual Waiting Time ◽  
Server Queue ◽  
Many Server Queues ◽  
The Many ◽  
Busy Cycle

Pollaczek's theory for the many server queue is generalized and extended. Pollaczek (1961) found the distribution of the actual waiting times in the model G/G/s as a solution of a set of integral equations. We give a somewhat more general set of integral equations from which the joint distribution of the actual waiting time and some other random variables may be found. With this joint distribution we can obtain distributions of a number of characteristic quantities, such as the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. For a wide class of many server queues the formal expressions may lead to explicit results.

On the virtual and actual waiting time distributions of a GI/G/1 queue

1976 ◽  
Vol 13 (4) ◽  
pp. 833-836 ◽  
Author(s):  
J. Michael Harrison ◽  
Austin J. Lemoine
Keyword(s):  
Waiting Time ◽  
Sample Path ◽  
Interarrival Time ◽  
Waiting Time Distributions ◽  
Poisson Input ◽  
Imbedded Markov Chain ◽  
Virtual Waiting Time ◽  
Similar Proof ◽  
The Relationship ◽  
Special Case

Consider a stable GI/G/1 queue with non-lattice interarrival time distribution. Let G and H be the limiting actual and virtual waiting time distributions respectively. Two separate statements of the relationship between G and H are found in a classical theorem of Takàcs and a more recent (and previously unpublished) theorem of Hooke. A simplified proof of Takàcs's theorem, based on a sample path relationship between the virtual and actual waiting time processes, has recently been advanced. This paper gives a similar proof of Hooke's theorem, based on the same sample path relationship, and demonstrates the utility of the result in analyzing the special case of Poisson input. In particular, by combining the Takàcs and Hooke results one can obtain the Pollaczek–Khintchine formula without any reference to the imbedded Markov chain.

The queue GI/Hm/s in continuous time

1985 ◽  
Vol 22 (01) ◽  
pp. 214-222 ◽  
Author(s):  
Jos H. A. De smit
Keyword(s):  
Waiting Time ◽  
Continuous Time ◽  
Virtual Waiting Time ◽  
Number Of Customers ◽  
Busy Cycle

This is an extension of our previous paper [4] on the queue GI/Hm/s. In that paper we have derived results for the actual waiting time, the number of customers in the system at arrival epochs and the number of customers during a busy cycle. Here we obtain results for the virtual waiting time and the number of customers in the system at arbitrary times.

Generalizations of the Pollaczek-Khinchin integral equation in the theory of queues

1986 ◽  
Vol 18 (4) ◽  
pp. 952-990 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Waiting Time ◽  
Volterra Integral Equations ◽  
Markov Renewal Process ◽  
Waiting Time Distributions ◽  
Virtual Waiting Time ◽  
Bulk Service ◽  
Constant Coefficients ◽  
Linear Differential

A classical result in queueing theory states that in the stable M/G/1 queue, the stationary distribution W(x) of the waiting time of an arriving customer or of the virtual waiting time satisfies a linear Volterra integral equation of the second kind, of convolution type. For many variants of the M/G/1 queue, there are corresponding integral equations, which in most cases differ from the Pollaczek–Khinchin equation only in the form of the inhomogeneous term. This leads to interesting factorizations of the waiting-time distribution and to substantial algorithmic simplifications. In a number of priority queues, the waiting-time distributions satisfy Volterra integral equations whose kernel is a functional of the busy-period distribution in related M/G/1 queues. In other models, such as the M/G/1 queue with Bernoulli feedback or with limited admissions of customers per service, there is a more basic integral equation of Volterra type, which yields a probability distribution in terms of which the waiting-time distributions are conveniently expressed.For several complex queueing models with an embedded Markov renewal process of M/G/1 type, one obtains matrix Volterra integral equations for the waiting-time distributions or for related vectors of mass functions. Such models include the M/SM/1 and the N/G/1 queues, as well as the M/G/1 queue with some forms of bulk service.When the service-time distributions are of phase type, the numerical computation of waiting-time distributions may commonly be reduced to the solution of systems of linear differential equations with constant coefficients.

Sign in / Sign up

Export Citation Format

Share Document