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

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.

1972 ◽  
Vol 9 (01) ◽  
pp. 178-184 ◽  
Author(s):  
Marcel F. Neuts ◽  
Shun-Zer Chen

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.


1999 ◽  
Vol 12 (1) ◽  
pp. 35-62 ◽  
Author(s):  
Doo Il Choi ◽  
Charles Knessl ◽  
Charles Tier

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.


1973 ◽  
Vol 5 (1) ◽  
pp. 170-182 ◽  
Author(s):  
J. H. A. De Smit

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.


1985 ◽  
Vol 22 (04) ◽  
pp. 893-902 ◽  
Author(s):  
Hermann Thorisson

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.


1973 ◽  
Vol 5 (01) ◽  
pp. 170-182 ◽  
Author(s):  
J. H. A. De Smit

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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