The M/G/k group-arrival group-departure loss system

1982 ◽  
Vol 19 (4) ◽  
pp. 826-834 ◽  
Author(s):  
D. Fakinos
Keyword(s):  
Poisson Process ◽  
Group Size ◽  
Loss System ◽  
Group Arrival ◽  
Service Times ◽  
Departure Processes

This paper considers the equilibrium behaviour of the M/G/k group-arrival group-departure loss system. Such a system has k servers whose customers arrive in groups, the arrival epochs of groups being points of a Poisson process. The duration of a service can be characteristic of the group size; however, customers who belong to the same group have equal service times. The customers of a group start being served immediately upon their arrival, unless their number is greater than the number of idle servers. In this case the whole group leaves and does not return later (i.e. is lost). Among other things, a generalization of the Erlang B-formula is given and it is shown that the arrival and departure processes are statistically indistinguishable.

The M/G/k group-arrival group-departure loss system

1982 ◽  
Vol 19 (04) ◽  
pp. 826-834 ◽  
Author(s):  
D. Fakinos
Keyword(s):  
Poisson Process ◽  
Group Size ◽  
Loss System ◽  
Group Arrival ◽  
Service Times ◽  
Departure Processes

This paper considers the equilibrium behaviour of the M/G/k group-arrival group-departure loss system. Such a system has k servers whose customers arrive in groups, the arrival epochs of groups being points of a Poisson process. The duration of a service can be characteristic of the group size; however, customers who belong to the same group have equal service times. The customers of a group start being served immediately upon their arrival, unless their number is greater than the number of idle servers. In this case the whole group leaves and does not return later (i.e. is lost). Among other things, a generalization of the Erlang B-formula is given and it is shown that the arrival and departure processes are statistically indistinguishable.

Stationary Poisson departure processes from non-stationary queues

1986 ◽  
Vol 23 (1) ◽  
pp. 256-260 ◽  
Author(s):  
Robert D. Foley
Keyword(s):  
Poisson Process ◽  
Service Time ◽  
Queueing Systems ◽  
Arrival Rate ◽  
Distribution Functions ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Departure Process ◽  
Infinite Server ◽  
Departure Processes

We present some non-stationary infinite-server queueing systems with stationary Poisson departure processes. In Foley (1982), it was shown that the departure process from the Mt/Gt/∞ queue was a Poisson process, possibly non-stationary. The Mt/Gt/∞ queue is an infinite-server queue with a stationary or non-stationary Poisson arrival process and a general server in which the service time of a customer may depend upon the customer's arrival time. Mirasol (1963) pointed out that the departure process from the M/G/∞ queue is a stationary Poisson process. The question arose whether there are any other Mt/Gt/∞ queueing systems with stationary Poisson departure processes. For example, if the arrival rate is periodic, is it possible to select the service-time distribution functions to fluctuate in order to compensate for the fluctuations of the arrival rate? In this situation and in more general situations, it is possible to select the server such that the system yields a stationary Poisson departure process.

A service model in which the server is required to search for customers

1984 ◽  
Vol 21 (1) ◽  
pp. 157-166 ◽  
Author(s):  
Marcel F. Neuts ◽  
M. F. Ramalhoto
Keyword(s):  
Poisson Process ◽  
Numerical Computation ◽  
Probability Distributions ◽  
General Distribution ◽  
Search Process ◽  
Single Server ◽  
Stationary Probability ◽  
Service Model ◽  
Service Times ◽  
Number Of Customers

Customers enter a pool according to a Poisson process and wait there to be found and processed by a single server. The service times of successive items are independent and have a common general distribution. Successive services are separated by seek phases during which the server searches for the next customer. The search process is Markovian and the probability of locating a customer in (t, t + dt) is proportional to the number of customers in the pool at time t. Various stationary probability distributions for this model are obtained in explicit forms well-suited for numerical computation.Under the assumption of exponential service times, corresponding results are obtained for the case where customers may escape from the pool.

scholarly journals Product-form distributions for an M/G/k group-arrival group-departure loss system

1989 ◽  
Vol 21 (3) ◽  
pp. 721-724 ◽  
Author(s):  
D. Fakinos ◽  
K. Sirakoulis
Keyword(s):  
Equilibrium Distribution ◽  
Product Form ◽  
Loss System ◽  
Group Arrival

In this work we investigate under what circumstances the equilibrium distribution of the numbers of groups of various sizes in a certain M/G/k group-arrival group-departure loss system can be obtained in a closed product form.

A note on Belyaev's limiting distribution of the intervals between losses in an n-server system

1970 ◽  
Vol 7 (2) ◽  
pp. 457-464 ◽  
Author(s):  
D. G. Tambouratzis
Keyword(s):  
Exponential Distribution ◽  
Present Note ◽  
Limiting Distribution ◽  
Loss System ◽  
Negative Exponential Distribution ◽  
Service Times ◽  
Negative Exponential ◽  
Server System

SummaryThe aim of the present note is to give an alternative simpler proof to a result of Belyaev [1], namely that in a loss system of n servers with recurrent input and negative exponential service times the intervals between losses, suitably scaled to have constant mean, tend to a negative exponential distribution as n tends to infinity.

The M/G/k loss system with servers subject to breakdown

1983 ◽  
Vol 20 (3) ◽  
pp. 706-712 ◽  
Author(s):  
D. Fakinos
Keyword(s):  
Poisson Process ◽  
Loss System ◽  
Departure Process ◽  
Probability Of Occurrence ◽  
State Dependent

This paper studies the equilibrium behaviour of the M/G/k loss system with servers subject to breakdown. Such a system has k servers, whose customers arrive in a Poisson process. They are served if there is an idle server, otherwise they leave and do not return. Each server is subject to breakdown with probability of occurrence depending on the length of the time the server has been busy since his last repair. On a breakdown the customer waits and his service is continued just after the repair of the server. Among other things, a generalization of the Erlang B-formula is given and it is shown that the equilibrium departure process is Poisson. In fact these results are obtained for the more general case where customers may balk and service and repair rates are state dependent.

Equilibrium results for the M/G/k group-arrival loss system

Top ◽  
2010 ◽  
Vol 21 (1) ◽  
pp. 163-181
Author(s):  
Spiros Dimou ◽  
Demetrios Fakinos
Keyword(s):  
Loss System ◽  
Group Arrival

scholarly journals A dam with general release rule

1976 ◽  
Vol 19 (4) ◽  
pp. 469-477 ◽  
Author(s):  
Geoffrey Yeo
Keyword(s):  
Poisson Process ◽  
Waiting Time ◽  
Release Rate ◽  
Renewal Process ◽  
Content Distribution ◽  
Size Distributions ◽  
Release Rates ◽  
Input Size ◽  
Service Times

AbstractA dam is considered with independently and identically distributed inputs occurring in a renewal process, and in particular a Poisson process, with a general release rate r(·) depending on the content. This is related to a GI/G/1 queue with service times dependent on the waiting time. Some results are obtained for the limiting content distribution when it exists; these are more complete for some special release rates, such as r(x) = μxα and r(x) = a + μx, and particular input size distributions.

scholarly journals Stability of retrial queues with versatile retrial policy

2006 ◽  
Vol 2006 ◽  
pp. 1-16 ◽  
Author(s):  
Tewfik Kernane ◽  
Amar Aïssani
Keyword(s):  
Poisson Process ◽  
Strong Coupling ◽  
Sufficient Conditions ◽  
Retrial Queues ◽  
Negative Customers ◽  
Batch Arrivals ◽  
Unreliable Server ◽  
Service Times ◽  
The Stability

We consider in this paper the stability of retrial queues with a versatile retrial policy. We obtain sufficient conditions for the stability by the strong coupling convergence to a stationary ergodic regime for various models of retrial queues including a model with two types of customers, a model with breakdowns of the server, a model with negative customers, and a model with batch arrivals. For all the models considered we assume that the service times are general stationary ergodic and interarrival and retrial times are i.i.d. sequences exponentially distributed. For the model with unreliable server we also assume that the repair times are stationary and ergodic and the occurrences of breakdowns follow a Poisson process.

A Queueing Loss Model with Heterogeneous Skill Based Servers Under Idle Time Ordering Policies

2015 ◽  
Vol 52 (01) ◽  
pp. 269-277 ◽  
Author(s):  
Babak Haji ◽  
Sheldon M. Ross
Keyword(s):  
Service Time ◽  
Idle Time ◽  
Limiting Distribution ◽  
Loss System ◽  
Loss Model ◽  
Poisson Arrivals ◽  
Service Times ◽  
Ordering Policies

We consider a queueing loss system with heterogeneous skill based servers with arbitrary service distributions. We assume Poisson arrivals, with each arrival having a vector indicating which of the servers are eligible to serve it. An arrival can only be assigned to a server that is both idle and eligible. Assuming exchangeable eligibility vectors and an idle time ordering assignment policy, the limiting distribution of the system is derived. It is shown that the limiting probabilities of the set of idle servers depend on the service time distributions only through their means. Moreover, conditional on the set of idle servers, the remaining service times of the busy servers are independent and have their respective equilibrium service distributions.

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