The infinite server queue with semi-Markovian arrivals and negative exponential services

1972 ◽  
Vol 9 (01) ◽  
pp. 178-184 ◽  
Author(s):  
Marcel F. Neuts ◽  
Shun-Zer Chen
Keyword(s):  
Discrete Time ◽  
Infinite Number ◽  
Queue Length ◽  
Markovian Arrival Process ◽  
Arrival Process ◽  
Infinite Server ◽  
Server Queue ◽  
Service Times ◽  
Negative Exponential ◽  
Stationary Probabilities

The queue with an infinite number of servers with a semi-Markovian arrival process and with negative exponential service times is studied. The queue length process and the type of the last customer to join the queue before time t are studied jointly, both in continuous and in discrete time. Limiting stationary probabilities are also obtained.

The infinite server queue with semi-Markovian arrivals and negative exponential services

1972 ◽  
Vol 9 (1) ◽  
pp. 178-184 ◽  
Author(s):  
Marcel F. Neuts ◽  
Shun-Zer Chen
Keyword(s):  
Discrete Time ◽  
Infinite Number ◽  
Queue Length ◽  
Markovian Arrival Process ◽  
Arrival Process ◽  
Infinite Server ◽  
Server Queue ◽  
Service Times ◽  
Negative Exponential ◽  
Stationary Probabilities

The queue with an infinite number of servers with a semi-Markovian arrival process and with negative exponential service times is studied. The queue length process and the type of the last customer to join the queue before time t are studied jointly, both in continuous and in discrete time. Limiting stationary probabilities are also obtained.

scholarly journals A queueing system with queue length dependent service times, with applications to cell discarding in ATM networks

1999 ◽  
Vol 12 (1) ◽  
pp. 35-62 ◽  
Author(s):  
Doo Il Choi ◽  
Charles Knessl ◽  
Charles Tier
Keyword(s):  
Service Time ◽  
Queue Length ◽  
Queueing System ◽  
Arrival Process ◽  
Tail Probabilities ◽  
Asymptotic Results ◽  
Time Service ◽  
Service Times ◽  
General Service ◽  
Stationary Probabilities

A queueing system (M/G1,G2/1/K) is considered in which the service time of a customer entering service depends on whether the queue length, N(t), is above or below a threshold L. The arrival process is Poisson, and the general service times S1 and S2 depend on whether the queue length at the time service is initiated is <L or ≥L, respectively. Balance equations are given for the stationary probabilities of the Markov process (N(t),X(t)), where X(t) is the remaining service time of the customer currently in service. Exact solutions for the stationary probabilities are constructed for both infinite and finite capacity systems. Asymptotic approximations of the solutions are given, which yield simple formulas for performance measures such as loss rates and tail probabilities. The numerical accuracy of the asymptotic results is tested.

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 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.

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.

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