Analysis of two parallel queues by common exit service and infinite queue size

1981 ◽  
Vol 13 (1) ◽  
pp. 147-166
Author(s):  
C. Atkinson ◽  
M. E. Thompson
Keyword(s):  
Integral Equations ◽  
Service Time ◽  
Queue Size ◽  
Parallel Queues ◽  
Fixed Length ◽  
Common Gate ◽  
Erlangian Service Times ◽  
Poisson Arrivals ◽  
Service Times ◽  
Independent And Identically Distributed

A system of two parallel queues is considered, where each customer must leave after service through a common gate G. It is assumed that service times at the two stations I and II are independent and identically distributed, and that exit service takes a fixed length of time. A I-customer may be served at station I only if the previous I-customer has completed exit service. Integral equations are formulated from which the distribution of the total service time may be obtained when the two queue sizes are infinite. These equations are solved for exponential and generalized erlangian service times. Extensions to the case of k parallel queues and to the case of Poisson arrivals and finite queue sizes are discussed briefly.

Analysis of two parallel queues by common exit service and infinite queue size

1981 ◽  
Vol 13 (01) ◽  
pp. 147-166
Author(s):  
C. Atkinson ◽  
M. E. Thompson
Keyword(s):  
Integral Equations ◽  
Service Time ◽  
Queue Size ◽  
Parallel Queues ◽  
Fixed Length ◽  
Common Gate ◽  
Erlangian Service Times ◽  
Poisson Arrivals ◽  
Service Times ◽  
Independent And Identically Distributed

A system of two parallel queues is considered, where each customer must leave after service through a common gate G. It is assumed that service times at the two stations I and II are independent and identically distributed, and that exit service takes a fixed length of time. A I-customer may be served at station I only if the previous I-customer has completed exit service. Integral equations are formulated from which the distribution of the total service time may be obtained when the two queue sizes are infinite. These equations are solved for exponential and generalized erlangian service times. Extensions to the case of k parallel queues and to the case of Poisson arrivals and finite queue sizes are discussed briefly.

Queues with moving average service times

1967 ◽  
Vol 4 (03) ◽  
pp. 553-570 ◽  
Author(s):  
C. Pearce
Keyword(s):  
Service Time ◽  
Moving Average ◽  
Length Distribution ◽  
Transient State ◽  
Single Server ◽  
Queue Length Distribution ◽  
Mean And Variance ◽  
Poisson Arrivals ◽  
The Mean ◽  
Service Times

A model for the service time structure in the single server queue is given embodying correlations between contiguous and near-contiguous service times. A number of results are derived in the case of Poisson arrivals both for equilibrium and the transient state. In particular, Kendall's (equilibrium) result P (a departure leaves the queue empty) = 1 — (mean service time)/(mean inter-arrival time) is found still to hold good. The effect of the correlation on the mean and variance of the equilibrium queue length distribution is examined in a simple case.

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.

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

2015 ◽  
Vol 52 (1) ◽  
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.

Queues with moving average service times

1967 ◽  
Vol 4 (3) ◽  
pp. 553-570 ◽  
Author(s):  
C. Pearce
Keyword(s):  
Service Time ◽  
Moving Average ◽  
Length Distribution ◽  
Transient State ◽  
Single Server ◽  
Queue Length Distribution ◽  
Mean And Variance ◽  
Poisson Arrivals ◽  
The Mean ◽  
Service Times

A model for the service time structure in the single server queue is given embodying correlations between contiguous and near-contiguous service times. A number of results are derived in the case of Poisson arrivals both for equilibrium and the transient state. In particular, Kendall's (equilibrium) result P (a departure leaves the queue empty) = 1 — (mean service time)/(mean inter-arrival time) is found still to hold good. The effect of the correlation on the mean and variance of the equilibrium queue length distribution is examined in a simple case.

The single-server queue with service depending on queue size and with the preemptive-resume last-come–first-served queue discipline

1987 ◽  
Vol 24 (03) ◽  
pp. 758-767
Author(s):  
D. Fakinos
Keyword(s):  
Queueing System ◽  
Limiting Distribution ◽  
Single Server ◽  
Queue Size ◽  
Single Server Queue ◽  
Preemptive Resume ◽  
Server Queue ◽  
Queue Discipline ◽  
Service Times

This paper studies theGI/G/1 queueing system assuming that customers have service times depending on the queue size and also that they are served in accordance with the preemptive-resume last-come–first-served queue discipline. Expressions are given for the limiting distribution of the queue size and the remaining durations of the corresponding services, when the system is considered at arrival epochs, at departure epochs and continuously in time. Also these results are applied to some particular cases of the above queueing system.

scholarly journals Tails in generalized Jackson networks with subexponential service-time distributions

2005 ◽  
Vol 42 (2) ◽  
pp. 513-530 ◽  
Author(s):  
François Baccelli ◽  
Serguei Foss ◽  
Marc Lelarge
Keyword(s):  
Closed Form ◽  
Service Time ◽  
Large Deviation ◽  
Exact Asymptotics ◽  
Fluid Limits ◽  
Jackson Networks ◽  
Single Station ◽  
Service Times

We give the exact asymptotics of the tail of the stationary maximal dater in generalized Jackson networks with subexponential service times. This maximal dater, which is an analogue of the workload in an isolated queue, gives the time taken to clear all customers present at some time t when stopping all arrivals that take place later than t. We use the property that a large deviation of the maximal dater is caused by a single large service time at a single station at some time in the distant past of t, in conjunction with fluid limits of generalized Jackson networks, to derive the relevant asymptotics in closed form.

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.

Extremal properties of the FIFO discipline in queueing networks

1992 ◽  
Vol 29 (4) ◽  
pp. 967-978 ◽  
Author(s):  
Rhonda Righter ◽  
J. George Shanthikumar
Keyword(s):  
Likelihood Ratio ◽  
Service Time ◽  
Queueing Networks ◽  
Service Rate ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queues ◽  
First Results ◽  
Service Times ◽  
Number Of Customers

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

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.

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