The queue GI/M/s with customers of different types or the queue GI/Hm/s

1983 ◽  
Vol 15 (2) ◽  
pp. 392-419 ◽  
Author(s):  
Jos H. A. De Smit
Keyword(s):  
Service Time ◽  
Waiting Times ◽  
Factorization Method ◽  
Exponential Distributions ◽  
Probabilistic Argument ◽  
Stationary Distributions ◽  
Different Types ◽  
Queue Lengths

We study the queue GI/M/s with customers of m different types. An arriving customer is of type i with probability pi and the types of different customers are independent. A customer of type i requires a service time which is exponentially distributed with parameter bi. This model is equivalent to the queue GI/Hm/s, where Hm denotes a mixture of m different exponential distributions. We are primarily interested in the distributions of waiting times and queue lengths. Using a probabilistic argument we reduce the problem to the solution of a system of Wiener-Hopf-type equations. This system is solved by a factorization method. Thus we obtain explicit results for the stationary distributions of waiting times and queue lengths.

The queue GI/M/s with customers of different types or the queue GI/Hm/s

1983 ◽  
Vol 15 (02) ◽  
pp. 392-419 ◽  
Author(s):  
Jos H. A. De Smit
Keyword(s):  
Service Time ◽  
Waiting Times ◽  
Factorization Method ◽  
Exponential Distributions ◽  
Probabilistic Argument ◽  
Stationary Distributions ◽  
Different Types ◽  
Queue Lengths

We study the queue GI/M/s with customers of m different types. An arriving customer is of type i with probability pi and the types of different customers are independent. A customer of type i requires a service time which is exponentially distributed with parameter bi . This model is equivalent to the queue GI/Hm/s, where Hm denotes a mixture of m different exponential distributions. We are primarily interested in the distributions of waiting times and queue lengths. Using a probabilistic argument we reduce the problem to the solution of a system of Wiener-Hopf-type equations. This system is solved by a factorization method. Thus we obtain explicit results for the stationary distributions of waiting times and queue lengths.

The probability of large queue lengths and waiting times in a heterogeneous multiserver queue I: Tight limits

1995 ◽  
Vol 27 (02) ◽  
pp. 532-566 ◽  
Author(s):  
John S. Sadowsky ◽  
Wojciech Szpankowski
Keyword(s):  
Service Time ◽  
Waiting Times ◽  
Renewal Theory ◽  
Batch Size ◽  
Interarrival Time ◽  
Recurrent Markov Chain ◽  
Queue Lengths ◽  
Exponential Change ◽  
Stationary Probabilities

We consider a multiserver queuing process specified by i.i.d. interarrival time, batch size and service time sequences. In the case that different servers have different service time distributions we say the system is heterogeneous. In this paper we establish conditions for the queuing process to be characterized as a geometrically Harris recurrent Markov chain, and we characterize the stationary probabilities of large queue lengths and waiting times. The queue length is asymptotically geometric and the waiting time is asymptotically exponential. Our analysis is a generalization of the well-known characterization of the GI/G/1 queue obtained using classical probabilistic techniques of exponential change of measure and renewal theory.

Queues with marked customers

1996 ◽  
Vol 28 (02) ◽  
pp. 567-587 ◽  
Author(s):  
Qi-Ming He
Keyword(s):  
Detailed Analysis ◽  
Waiting Times ◽  
Queueing Systems ◽  
Fundamental Period ◽  
Different Types ◽  
Arrival Processes ◽  
Queue Lengths ◽  
Points Of View ◽  
Markov Arrival

Queueing systems with distinguished arrivals are described on the basis of Markov arrival processes with marked transitions. Customers are distinguished by their types of arrival. Usually, the queues observed by customers of different types are different, especially for queueing systems with bursty arrival processes. We study queueing systems from the points of view of customers of different types. A detailed analysis of the fundamental period, queue lengths and waiting times at the epochs of arrivals is given. The results obtained are the generalizations of the results of theMAP/G/1 queue.

A service system with two stages of waiting and feedback of customers

1984 ◽  
Vol 21 (02) ◽  
pp. 404-413 ◽  
Author(s):  
Osman M. E. Ali ◽  
Marcel F. Neuts
Keyword(s):  
Service System ◽  
Waiting Times ◽  
Random Numbers ◽  
Stationary Distributions ◽  
Waiting Room ◽  
Service Unit ◽  
Queue Lengths ◽  
Two Stages

Customers initially enter a service unit via a waiting room. The customers to be served are stored in a service room which is replenished by the transfer of all those in the waiting room at the points in time where the service room becomes empty. At those epochs of transfer, positive random numbers of ‘overhead customers' are also added to the service room. Algorithmically tractable expressions for the stationary distributions of queue lengths and waiting times at various embedded random epochs are derived. The discussion generalizes an earlier treatment by Takács in several directions.

A service system with two stages of waiting and feedback of customers

1984 ◽  
Vol 21 (2) ◽  
pp. 404-413 ◽  
Author(s):  
Osman M. E. Ali ◽  
Marcel F. Neuts
Keyword(s):  
Service System ◽  
Waiting Times ◽  
Random Numbers ◽  
Stationary Distributions ◽  
Waiting Room ◽  
Service Unit ◽  
Queue Lengths ◽  
Two Stages

Customers initially enter a service unit via a waiting room. The customers to be served are stored in a service room which is replenished by the transfer of all those in the waiting room at the points in time where the service room becomes empty. At those epochs of transfer, positive random numbers of ‘overhead customers' are also added to the service room. Algorithmically tractable expressions for the stationary distributions of queue lengths and waiting times at various embedded random epochs are derived. The discussion generalizes an earlier treatment by Takács in several directions.

Queues with marked customers

1996 ◽  
Vol 28 (2) ◽  
pp. 567-587 ◽  
Author(s):  
Qi-Ming He
Keyword(s):  
Detailed Analysis ◽  
Waiting Times ◽  
Queueing Systems ◽  
Fundamental Period ◽  
Different Types ◽  
Arrival Processes ◽  
Queue Lengths ◽  
Points Of View ◽  
Markov Arrival

Queueing systems with distinguished arrivals are described on the basis of Markov arrival processes with marked transitions. Customers are distinguished by their types of arrival. Usually, the queues observed by customers of different types are different, especially for queueing systems with bursty arrival processes. We study queueing systems from the points of view of customers of different types. A detailed analysis of the fundamental period, queue lengths and waiting times at the epochs of arrivals is given. The results obtained are the generalizations of the results of the MAP/G/1 queue.

The probability of large queue lengths and waiting times in a heterogeneous multiserver queue I: Tight limits

1995 ◽  
Vol 27 (2) ◽  
pp. 532-566 ◽  
Author(s):  
John S. Sadowsky ◽  
Wojciech Szpankowski
Keyword(s):  
Service Time ◽  
Waiting Times ◽  
Renewal Theory ◽  
Batch Size ◽  
Interarrival Time ◽  
Recurrent Markov Chain ◽  
Queue Lengths ◽  
Exponential Change ◽  
Stationary Probabilities

We consider a multiserver queuing process specified by i.i.d. interarrival time, batch size and service time sequences. In the case that different servers have different service time distributions we say the system is heterogeneous. In this paper we establish conditions for the queuing process to be characterized as a geometrically Harris recurrent Markov chain, and we characterize the stationary probabilities of large queue lengths and waiting times. The queue length is asymptotically geometric and the waiting time is asymptotically exponential. Our analysis is a generalization of the well-known characterization of the GI/G/1 queue obtained using classical probabilistic techniques of exponential change of measure and renewal theory.

On a tandem queueing model with identical service times at both counters, I

1979 ◽  
Vol 11 (3) ◽  
pp. 616-643 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Waiting Times ◽  
Sufficient Conditions ◽  
Complete Analysis ◽  
Single Server ◽  
Stationary Distributions ◽  
In Series ◽  
Queues In Series ◽  
Service Times ◽  
Necessary And Sufficient ◽  
Insight Into

This paper considers a queueing system consisting of two single-server queues in series, in which the service times of an arbitrary customer at both queues are identical. Customers arrive at the first queue according to a Poisson process.Of this model, which is of importance in modern network design, a rather complete analysis will be given. The results include necessary and sufficient conditions for stationarity of the tandem system, expressions for the joint stationary distributions of the actual waiting times at both queues and of the virtual waiting times at both queues, and explicit expressions (i.e., not in transform form) for the stationary distributions of the sojourn times and of the actual and virtual waiting times at the second queue.In Part II (pp. 644–659) these results will be used to obtain asymptotic and numerical results, which will provide more insight into the general phenomenon of tandem queueing with correlated service times at the consecutive queues.

On queueing systems by retrials

1983 ◽  
Vol 20 (02) ◽  
pp. 380-389 ◽  
Author(s):  
Vidyadhar G. Kulkarni
Keyword(s):  
General Result ◽  
Service Time ◽  
Queueing Systems ◽  
Single Server ◽  
Expected Number ◽  
Waiting Room ◽  
Second Moments ◽  
Different Types ◽  
General Distributions ◽  
Arbitrary Service Time

A general result for queueing systems with retrials is presented. This result relates the expected total number of retrials conducted by an arbitrary customer to the expected total number of retrials that take place during an arbitrary service time. This result is used in the analysis of a special system where two types of customer arrive in an independent Poisson fashion at a single-server service station with no waiting room. The service times of the two types of customer have independent general distributions with finite second moments. When the incoming customer finds the server busy he immediately leaves and tries his luck again after an exponential amount of time. The retrial rates are different for different types of customers. Expressions are derived for the expected number of retrial customers of each type.

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