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.

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.

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.

Some comparability results for waiting times in single- and many-server queues

1984 ◽  
Vol 21 (4) ◽  
pp. 887-900 ◽  
Author(s):  
D. J. Daley ◽  
T. Rolski
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Waiting Times ◽  
Queueing Systems ◽  
Random Variables ◽  
Interarrival Time ◽  
Sufficient Condition ◽  
Traffic Intensity ◽  
Many Server Queues ◽  
Stationary Waiting Time

It is shown that the stationary waiting time random variables W′, W″ of two M/G/l queueing systems for which the corresponding service time random variables satisfy E(S′−x)+ ≦ E(S″−x)+ (all x >0), are stochastically ordered as W′≦dW″. The weaker conclusion, that E(W′−x)+ ≦ E(W″−x)+ (all x > 0), is shown to hold in GI/M/k systems when the interarrival time random variables satisfy E(x−T′)+ ≦ E(x−T″)+ (all x). A sufficient condition for wk≡EW in GI/D/k to be monotonic in k for a sequence of k-server queues with the same relative traffic intensity is given. Evidence indicating or refuting possible strengthenings of some of the results is indicated.

A Markov chain approach to periodic queues

1987 ◽  
Vol 24 (1) ◽  
pp. 215-225 ◽  
Author(s):  
Søren Asmussen ◽  
Hermann Thorisson
Keyword(s):  
Markov Chain ◽  
Waiting Times ◽  
Suitable Condition ◽  
Time Dependent ◽  
Interarrival Time ◽  
Traffic Intensity ◽  
Markov Chain Approach ◽  
Periodic Queues ◽  
Queue Lengths ◽  
Dependent Processes

We consider GI/G/1 queues in an environment which is periodic in the sense that the service time of the nth customer and the next interarrival time depend on the phase θ n at the arrival instant. Assuming Harris ergodicity of {θ n} and a suitable condition on the traffic intensity, various Markov chains related to the queue are then again Harris ergodic and provide limit results for the standard customer- and time-dependent processes such as waiting times and queue lengths. As part of the analysis, a result of Nummelin (1979) concerning Lindley processes on a Markov chain is reconsidered.

A Markov chain approach to periodic queues

1987 ◽  
Vol 24 (01) ◽  
pp. 215-225 ◽  
Author(s):  
Søren Asmussen ◽  
Hermann Thorisson
Keyword(s):  
Markov Chain ◽  
Waiting Times ◽  
Suitable Condition ◽  
Time Dependent ◽  
Interarrival Time ◽  
Traffic Intensity ◽  
Markov Chain Approach ◽  
Periodic Queues ◽  
Queue Lengths ◽  
Dependent Processes

We consider GI/G/1 queues in an environment which is periodic in the sense that the service time of the nth customer and the next interarrival time depend on the phase θ n at the arrival instant. Assuming Harris ergodicity of {θ n } and a suitable condition on the traffic intensity, various Markov chains related to the queue are then again Harris ergodic and provide limit results for the standard customer- and time-dependent processes such as waiting times and queue lengths. As part of the analysis, a result of Nummelin (1979) concerning Lindley processes on a Markov chain is reconsidered.

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.

Some comparability results for waiting times in single- and many-server queues

1984 ◽  
Vol 21 (04) ◽  
pp. 887-900 ◽  
Author(s):  
D. J. Daley ◽  
T. Rolski
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Waiting Times ◽  
Queueing Systems ◽  
Random Variables ◽  
Interarrival Time ◽  
Sufficient Condition ◽  
Traffic Intensity ◽  
Many Server Queues ◽  
Stationary Waiting Time

It is shown that the stationary waiting time random variables W′, W″ of two M/G/l queueing systems for which the corresponding service time random variables satisfy E(S ′−x)+ ≦ E(S ″−x)+ (all x >0), are stochastically ordered as W ′≦d W ″. The weaker conclusion, that E(W ′−x)+ ≦ E(W ″−x)+ (all x > 0), is shown to hold in GI/M/k systems when the interarrival time random variables satisfy E(x−T ′)+ ≦ E(x−T ″)+ (all x). A sufficient condition for wk ≡EW in GI/D/k to be monotonic in k for a sequence of k-server queues with the same relative traffic intensity is given. Evidence indicating or refuting possible strengthenings of some of the results is indicated.

On a Markovian queue with weakly correlated interarrival times

1981 ◽  
Vol 18 (01) ◽  
pp. 190-203 ◽  
Author(s):  
Guy Latouche
Keyword(s):  
Time Distribution ◽  
Queueing System ◽  
Interarrival Time ◽  
Probability Vector ◽  
Common Value ◽  
Markovian Queue ◽  
The Stability ◽  
Number Of Customers ◽  
Stationary Probabilities ◽  
Stationary Probability Vector

A queueing system with exponential service and correlated arrivals is analysed. Each interarrival time is exponentially distributed. The parameter of the interarrival time distribution depends on the parameter for the preceding arrival, according to a Markov chain. The parameters of the interarrival time distributions are chosen to be equal to a common value plus a factor ofε, where ε is a small number. Successive arrivals are then weakly correlated. The stability condition is found and it is shown that the system has a stationary probability vector of matrix-geometric form. Furthermore, it is shown that the stationary probabilities for the number of customers in the system, are analytic functions ofε, for sufficiently smallε, and depend more on the variability in the interarrival time distribution, than on the correlations.

Characterization of Manufacturing System Computer Simulation using Taguchi Method

2015 ◽  
Vol 72 (4) ◽  
Author(s):  
Seyed Mojib Zahraee ◽  
Ali Chegeni ◽  
Jafri Mohd Rohani
Keyword(s):  
Computer Simulation ◽  
Taguchi Method ◽  
Manufacturing Industry ◽  
Waiting Times ◽  
Inspection Time ◽  
Total Output ◽  
Service Rate ◽  
Manufacturing Line

In the manufacturing industry, managers and engineers are trying to sustain their competitiveness by achieving high output and productivity. There are some common problems such as waiting times, failures, reworks in production line that impose extra cost to the companies. Therefore, companies are striving to find methods in order to determine and deal with problems using different methods such as mathematical, statistical and computer simulation. The goal of this paper is to increase the total output production and to improve productivity using computer simulation and Taguchi method. This paper introduces a color manufacturing line as a case study which is simulated using arena 13.9 software. Following that the Taguchi method is applied to assess the effect of controllable and uncontrollable factors on the total output production. According to the result of JMP 10 software to conduct Taguchi experiment, the maximum desirability of productivity will be achieved when the value of factors such service rate of delpak machine=UNIF (30, 40), number of labor=14, inspection time=120 and number of Permil=5. Taguchi Method plays an efficient and suitable role in the process improvement, proposing adjustments that will provide an improvement in the productivity. 

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