• The Korean Journal of Applied Statistics
• /
• v.24 no.2
• /
• pp.347-357
• /
• 2011

In this paper, we propose an approximation to the overshoot in M/$E_n$/1 queues. Overshoot means the size of excess over the threshold when the workload process of an M/$E_n$/1 queue exceeds a prespecified threshold. The distribution, $1^{st}$ and $2^{nd}$ moments of overshoot have an important role in solving some kind of optimization problems. For the approximation to the overshoot, we propose a formula that is a convex sum of the service time distribution and an exponential distribution. We also do a numerical study to check how exactly the proposed formula approximates the overshoot.

• Journal of applied mathematics & informatics
• /
• v.18 no.1_2
• /
• pp.431-439
• /
• 2005

The aim of this paper is to derive the analytical solution of the queue: $M/E_{r}/1/k/N$ for machine interference system with balking and reneging considering the discipline FIFO. Some measures of effectiveness and some special cases are obtained.

• The Korean Journal of Applied Statistics
• /
• v.28 no.1
• /
• pp.33-47
• /
• 2015

We consider an $M/E_n/1$ queueing model where customers arrive at a facility with a single server according to a Poisson process with customer service times assumed to be independent and identically distributed with Erlang distribution. We concentrate on the overshoot of the workload process in the queue. The overshoot means the excess over a threshold at the moment where the workload process exceeds the threshold. The approximation of the distribution of the overshoot was proposed by Bae et al. (2011); however, but the accuracy of the approximation was unsatisfactory. We derive an advanced approximation using the property of the Erlang distribution. Finally the newly proposed approximation is compared with the results of the previous study.

• Journal of applied mathematics & informatics
• /
• v.5 no.2
• /
• pp.403-426
• /
• 1998

We present an algorithm to find an approximation for the stationary distribution for the general ergodic spatially-inhomogeneous block-partitioned upper Hessenberg form. Our approximation makes use of an associated upper block-Hessenberg matrix which is spa-tially homogeneous except for a finite number of blocks. We treat the MAP/G/1 retrial queue and the retrial queue with two types of customer as specific instances and give some numerical examples. The numerical results suggest that our method is superior to the ordinary finite-truncation method.

• Communications of the Korean Mathematical Society
• /
• v.13 no.2
• /
• pp.393-403
• /
• 1998

We consider $M_1,M_2/M/1$ retrial queues with two classes of customers in which the service rates depend on the total number or the customers served since the beginning of the current busy period. In the case that arriving customers are bloced due to the channel being busy, the class 1 customers are queued in the priority group and are served as soon as the channel is free, whereas the class 2 customers enter the retrical group in order to try service again after a random amount of time. For the first $N(N \geq 1)$ exceptional services model which is a special case of our model, we derive the joint generating function of the numbers of customers in the two groups. When N = 1 i.e., the first exceptional service model, we obtain the joint generating function explicitly and if the arrival rate of class 2 customers is 0, we show that the results for our model coincide with known results for the M/M/1 queues with smart machine.

• Journal of Service Research and Studies
• /
• v.14 no.3
• /
• pp.172-187
• /
• 2024

This paper deals with asymptotic approximations analysis of M/M/s and M/D/s queues. For M/M/s queue, we observe "economies of scale" under the fixed utilization ρ and the fixed probability α that customer waits in system, how the average system size vary according to the number of servers s increasing. Simulation results show that as s increases, the number of servers who are idling increases, that is, the slack n-E[Q_n] diverges. In addition, through changing the waiting probability α under the M/M/s system, α was not highly sensitive to the behavior of the system size. And, it is shown that using ${\rho}_n\,=\,1-k/\sqrt{n}$ to handle heavy-traffic regime is only appropriate for k = 1 by observing the effect on the performance of the system with different values of k. For the M/D/s queue, two approximations are used to evaluate the expected system size under the fixed ρ and α. Simulations and comparison of these two approximations show that Cosmetatos' approximation performs quite well when the number of servers is small and traffic intensity is heavy, but it overestimates the true value for the large number of servers. Meanwhile, the modified approximation gives good results for the steady state count of the system although the number of servers grows large.

• Journal of Korean Institute of Industrial Engineers
• /
• v.27 no.1
• /
• pp.89-94
• /
• 2001

There are many queueing systems where customers wait for service up to a certain amount of time and leave the system if they are not served during that time. This paper considers a finite capacity multi-server queueing system with Poisson input and Erlang service time, where a customer becomes a lost customer when his service has not begun within an exponential patient time after his arrival. Performance measures such as average queue length, the average number of customers in service, and the proportion of lost customers can be obtained exactly through the proposed numerical solution procedure.