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.

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.

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.

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.

scholarly journals The Multiphase Queuing System of the Rijeka Airport

Pomorstvo ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 205-209
Author(s):  
Svjetlana Hess ◽  
Ana Grbčić
Keyword(s):  
Performance Indicators ◽  
Queueing Theory ◽  
Rare Case ◽  
Queueing System ◽  
Service System ◽  
Waiting Times ◽  
Real Problem ◽  
Single Server ◽  
Scientific Contribution ◽  
The Real

The paper gives an overview of the real system as a multiphase single server queuing problem, which is a rare case in papers dealing with the application of the queueing theory. The methodological and scientific contribution of this paper is primarily in setting up the model of the real problem applying the multiphase queueing theory. The research of service system at Rijeka Airport may allow the airport to be more competitive by increasing service quality. The existing performance measures have been evaluated in order to improve Rijeka Airport queueing system, as a record number of passengers is to be expected in the next few years. Performance indicators have pointed out how the system handles congestion. The research is also focused on defining potential bottlenecks and comparing the results with IATA guidelines in terms of maximum waiting times.

An M/G/1 model with finite waiting room in which a customer remains during part of service

1973 ◽  
Vol 10 (4) ◽  
pp. 778-785 ◽  
Author(s):  
Stig I. Rosenlund
Keyword(s):  
Service System ◽  
Asymptotic Expression ◽  
Length Distribution ◽  
Loss Ratio ◽  
Efficiency Measure ◽  
Waiting Room ◽  
System Of Integral Equations ◽  
Large Intensity ◽  
Long Run ◽  
Stieltjes Transforms

An M/G/1 service system with finite waiting room is studied. A customer is served by one server in phases, during some of which a place in the waiting room is occupied. The busy period length distribution is obtained from a system of integral equations leading to a linear system in Laplace-Stieltjes transforms. An asymptotic expression, for large intensity of arrival, for the expectation of this length is given. An efficiency measure giving the long run customer loss ratio is obtained. The model is shown to apply to an inventory and a container traffic problem.

The probability of large queue lengths and waiting times in a heterogeneous multiserver queue II: Positive recurrence and logarithmic limits

1995 ◽  
Vol 27 (2) ◽  
pp. 567-583 ◽  
Author(s):  
John S. Sadowsky
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Waiting Times ◽  
General Setting ◽  
Batch Arrival ◽  
Positive Recurrence ◽  
Waiting Time Distributions ◽  
Harris Recurrence ◽  
Stationary Queue Length ◽  
Queue Lengths

We continue our investigation of the batch arrival-heterogeneous multiserver queue begun in Part I. In a general setting we prove the positive Harris recurrence of the system, and with no additional conditions we prove logarithmic tail limits for the stationary queue length and waiting time distributions.

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.

Expected waiting times in a multiclass batch arrival retrial queue

1986 ◽  
Vol 23 (01) ◽  
pp. 144-154 ◽  
Author(s):  
V. G. Kulkarni
Keyword(s):  
Waiting Times ◽  
Retrial Queue ◽  
Batch Arrival ◽  
Single Server ◽  
Waiting Room ◽  
Compound Poisson ◽  
Special Cases

Expressions are derived for the expected waiting times for the customers of two types who arrive in batches (in a compound Poisson fashion) at a single-server queueing station with no waiting room. Those who cannot get served immediately keep returning to the system after random exponential amounts of time until they get served. The result is shown to agree with similar results for three special cases studied in the literature.

An M/G/1 model with finite waiting room in which a customer remains during part of service

1973 ◽  
Vol 10 (04) ◽  
pp. 778-785 ◽  
Author(s):  
Stig I. Rosenlund
Keyword(s):  
Service System ◽  
Asymptotic Expression ◽  
Length Distribution ◽  
Loss Ratio ◽  
Efficiency Measure ◽  
Waiting Room ◽  
System Of Integral Equations ◽  
Large Intensity ◽  
Long Run ◽  
Stieltjes Transforms

An M/G/1 service system with finite waiting room is studied. A customer is served by one server in phases, during some of which a place in the waiting room is occupied. The busy period length distribution is obtained from a system of integral equations leading to a linear system in Laplace-Stieltjes transforms. An asymptotic expression, for large intensity of arrival, for the expectation of this length is given. An efficiency measure giving the long run customer loss ratio is obtained. The model is shown to apply to an inventory and a container traffic problem.

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