On the many server queue with exponential service times

1973 ◽  
Vol 5 (1) ◽  
pp. 170-182 ◽  
Author(s):  
J. H. A. De Smit
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Busy Period ◽  
Virtual Waiting Time ◽  
Server Queue ◽  
Poisson Arrivals ◽  
Service Times ◽  
The Many ◽  
Special Case ◽  
Busy Cycle

The general theory for the many server queue due to Pollaczek (1961) and generalized by the author (de Smit (1973)) is applied to the system with exponential service times. In this way many explicit results are obtained for the distributions of characteristic quantities, such as the actual waiting time, the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. Most of these results are new, even for the special case of Poisson arrivals.

On the many server queue with exponential service times

1973 ◽  
Vol 5 (01) ◽  
pp. 170-182 ◽  
Author(s):  
J. H. A. De Smit
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Busy Period ◽  
Virtual Waiting Time ◽  
Server Queue ◽  
Poisson Arrivals ◽  
Service Times ◽  
The Many ◽  
Special Case ◽  
Busy Cycle

The general theory for the many server queue due to Pollaczek (1961) and generalized by the author (de Smit (1973)) is applied to the system with exponential service times. In this way many explicit results are obtained for the distributions of characteristic quantities, such as the actual waiting time, the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. Most of these results are new, even for the special case of Poisson arrivals.

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.

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.

A single server tandem queue

1971 ◽  
Vol 8 (1) ◽  
pp. 95-109 ◽  
Author(s):  
Sreekantan S. Nair
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Busy Period ◽  
Single Server ◽  
Queueing Model ◽  
Tandem Queue ◽  
Limiting Behavior ◽  
Virtual Waiting Time ◽  
Priority Model

Avi-Itzhak, Maxwell and Miller (1965) studied a queueing model with a single server serving two service units with alternating priority. Their model explored the possibility of having the alternating priority model treated in this paper with a single server serving alternately between two service units in tandem.Here we study the distribution of busy period, virtual waiting time and queue length and their limiting behavior.

On regenerative and ergodic properties of the k-server queue with non-stationary Poisson arrivals

1985 ◽  
Vol 22 (04) ◽  
pp. 893-902 ◽  
Author(s):  
Hermann Thorisson
Keyword(s):  
Queue Length ◽  
Rates Of Convergence ◽  
Regenerative Process ◽  
Stochastic Domination ◽  
Homogeneous Markov Process ◽  
Ergodic Properties ◽  
Server Queue ◽  
Poisson Arrivals ◽  
Service Times ◽  
Residual Service

We consider the stable k-server queue with non-stationary Poisson arrivals and i.i.d. service times and show that the non-time-homogeneous Markov process Zt = (the queue length and residual service times at time t) can be subordinated to a stable time-homogeneous regenerative process. As an application we show that if the system starts from given conditions at time s then the distribution of Zt stabilizes (but depends on t) as s tends backwards to –∞. Also moment and stochastic domination results are established for the delay and recurrence times of the regenerative process leading to results on uniform rates of convergence.

scholarly journals An M/G/1 queue with multiple types of feedback and gated vacations

1997 ◽  
Vol 34 (03) ◽  
pp. 773-784 ◽  
Author(s):  
Onno J. Boxma ◽  
Uri Yechiali
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queue Length ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Single Server Queue ◽  
Server Queue ◽  
Poisson Arrivals

This paper considers a single-server queue with Poisson arrivals and multiple customer feedbacks. If the first service attempt of a newly arriving customer is not successful, he returns to the end of the queue for another service attempt, with a different service time distribution. He keeps trying in this manner (as an ‘old' customer) until his service is successful. The server operates according to the ‘gated vacation' strategy; when it returns from a vacation to find K (new and old) customers, it renders a single service attempt to each of them and takes another vacation, etc. We study the joint queue length process of new and old customers, as well as the waiting time distribution of customers. Some extensions are also discussed.

On regenerative and ergodic properties of the k-server queue with non-stationary Poisson arrivals

1985 ◽  
Vol 22 (4) ◽  
pp. 893-902 ◽  
Author(s):  
Hermann Thorisson
Keyword(s):  
Queue Length ◽  
Rates Of Convergence ◽  
Regenerative Process ◽  
Stochastic Domination ◽  
Homogeneous Markov Process ◽  
Ergodic Properties ◽  
Server Queue ◽  
Poisson Arrivals ◽  
Service Times ◽  
Residual Service

We consider the stable k-server queue with non-stationary Poisson arrivals and i.i.d. service times and show that the non-time-homogeneous Markov process Zt = (the queue length and residual service times at time t) can be subordinated to a stable time-homogeneous regenerative process. As an application we show that if the system starts from given conditions at time s then the distribution of Zt stabilizes (but depends on t) as s tends backwards to –∞. Also moment and stochastic domination results are established for the delay and recurrence times of the regenerative process leading to results on uniform rates of convergence.

A single server tandem queue

1971 ◽  
Vol 8 (01) ◽  
pp. 95-109
Author(s):  
Sreekantan S. Nair
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Busy Period ◽  
Single Server ◽  
Queueing Model ◽  
Tandem Queue ◽  
Limiting Behavior ◽  
Virtual Waiting Time ◽  
Priority Model

Avi-Itzhak, Maxwell and Miller (1965) studied a queueing model with a single server serving two service units with alternating priority. Their model explored the possibility of having the alternating priority model treated in this paper with a single server serving alternately between two service units in tandem. Here we study the distribution of busy period, virtual waiting time and queue length and their limiting behavior.

A single-server queue with limited virtual waiting time

1974 ◽  
Vol 11 (03) ◽  
pp. 612-617 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Waiting Time ◽  
Single Server ◽  
Single Server Queue ◽  
Limiting Distributions ◽  
Poisson Input ◽  
Virtual Waiting Time ◽  
General Service Times ◽  
Server Queue ◽  
Service Times ◽  
General Service

The limiting distributions of the actual waiting time and the virtual waiting time are determined for a single-server queue with Poisson input and general service times in the case where there are two types of services and no customer can stay in the system longer than an interval of length m.

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