Queueing Loss Models

WB06.1 Asymptotic Behaviour of a Large Multi-Class Erlang Loss System

• Christine Fricker; INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt BP 105, Le Chesnay, 78153 , France; christine.fricker@inria.fr
• Philippe S. Robert; INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt BP 105, Le Chesnay, 78153 , France; philippe.robert@inria.fr
• Danielle Tibi; University Paris 7, Mathematiques Case 7012, 2 Plca Jussieu, Paris, , France; danielle.tibi@gauss.math.jussieu.fr

We consider the multi-class M/M/N/N queue. It is a single service station with N servers and no waiting room. Arrival processes are Poisson and service times are exponential. We study the asymptotic behaviour when the arrival rates of customers of each class and capacity of the station both become large, their ratio remains constant. The limit of the classic renormalized state process under the overcritical, undercritical and critical regimes is derived. For this particular loss system under the overcritical and critical regimes, a Skorohod reflection problem representation proves the functional law of large numbers derived by Hunt & Kurtz. Using the same technique and classic martingales convergence theorems, a central limit theorem under these 2 regimes is obtained starting for a point on the boundary in the attractive domain of the fixed point.

WB06.2 Asymptotic Analysis & Simple Approximation of the Loss Probability of the GI/M/c/K Queue with Group Arrivals

• Bara Kim; Korea University, Dept. of Math., 1, Anam-dong, Sungbuk-ku, Seoul, 136-701 , Korea; bara@semi.korea.ac.kr
• Bong D. Choi; Korea University, Dept. of Math., 1, Anam-dong, Sungbuk-ku, Seoul, 136-701 , Korea; bdchoi@semi.korea.ac.kr
• Jeongsim Kim; Korea University, Dept. of Math., 1, Anam-dong, Sungbuk-ku, Seoul, 136-701 , Korea; kjs@semi.korea.ac.kr

We consider the GI/M/c/K queue in which customers arrive in group and the system capacity including the service positions is limited to K. Recently, Laxmi & Gupta (Queueing Systems 2000) analyzed the GI/M/c/K queue with group arrivals through a combination of the supplementary variable method and the embedded Markov chain techniques. But, their analysis requires a lot of computational works for solving a system of linear equations with K+1 unknowns when K is large. Choi, Kim & Wee (Queueing Systems 2000) gave asymptotic loss probability of the GI/M/1/K queue as K tends to infinity.

We extend Choi's result to group arrival and multi-server system. Specifically, we give an asymptotic behavior of loss probability of the GI/M/c/K queue with group arrivals as K tends to infinity. The asymptotic loss probability is expressed only in terms of the roots of the characteristic equation and the boundary probabilities of the corresponding infinite queue. The asymptotic loss probability is used as an approximation of the loss probability of the GI(X)/M/c/K queue. The merits of this approximation of the loss probability are computational works are very small compared to the exact computation when K is large, the approximation is asymptotically correct as K tends to infinity and the approximate loss probability is obtained as a function of K, and so is very useful to find out proper buffer capacity K guaranteeing pre-specified loss probability. We give numerical examples in order to illustrate how well the asymptotic loss probability of the GI/M/c/K queue with group arrivals can be used as an approximation of the loss probability.

WB06.3 A MAP/M/c Queue with Constant Delay Bound

• Bong D. Choi; Korea University, Dept. of Math., 1, Anam-dong, Sungbuk-ku, Seoul, 136-701 , Korea; bdchoi@semi.korea.ac.kr
• Dong B. Zhu; KAIST, Div. of Applied Math., 373-1 Kusong-Dong, Yusong-Gu, Taejon, 305-701 , Korea; zhu@math.kaist.ac.kr
• Bara Kim; Korea University, Dept. of Math., 1, Anam-dong, Sungbuk-ku, Seoul, 136-701 , Korea; bara@semi.korea.ac.kr

We consider a MAP/M/c queue with constant delay bound. Customers arrive according to a Markovian arrival process (MAP). MAP is a fairly general process and is an appropriate model of arrivals in telecommunication systems. A customer who cannot begin to receive his service for a fixed time D becomes lost. We deal with two different cases of customer impatience: 'aware' customers and 'unaware' customers. In the case of aware customers, the service time of a customer is known upon arrival and the customer is rejected right away, if the workload of the server who has the minimal workload among all servers exceeds the queueing bound. In case of unaware customers, a customer always enters the system and is lost after waiting in the system, if he does not begin to receive service till the queueing bound.

Recently Choi et. al. (Queueing Systems 2000) and Brandt & Brandt (Method. Comp. Appl. Prob. 1999) dealt with 2-queue priority systems with impatient customers. We obtain the loss probability, waiting time distribution and system size distribution by finding and analyzing simple Markov process as follows: let N(t) be the system size at time t, and J(t) be the underlying Markov process of the MAP. Let V(t) is the virtual waiting time at time t which is defined as the waiting time of the virtual customer arrived at time t who has no bound of queueing time. It is observed that {(min{N(t),c},V(t),J(t))} is a Markov process. We obtain the stationary distribution of the Markov process by finding the density of the distribution as the form p(x)= c exp(A x)+ d exp(B t), 0

WB06.4 An Approximation of the Blocking Probability for M/G/C/C/+ N Queueing Systems

• J. MacGregor Smith; University of Massachusetts, Dept. of MIE, Amherst, MA 01003; jmsmith@ecs.umass.edu

Exact solutions for M/G/C/C + N queueing systems are only possible for special cases, such as exponential service, a single server or no waiting room. Instead of basing an approximation on the infinite capacity queue, as is often the case, an approximation is developed from the closed form expressions derivable from finite capacity systems. The performance of the approximation is compared with simulation models of the same systems along with demonstrations of the formula to various multi-server applications in manufacturing and service sector systems.