Bazı kuyruk modellerinde maliyet analizi ve simülasyonun rolü

Abstract

ÖZET Çoğu problemde olduğu gibi, kuyruk problemlerinin analizinde de maliyet oldukça önemli bir yer tutar. Söz konusu probleme ışık tutabilme amacıyla bu çalışmada, çe şitli kuyruk problemleri için maliyet modelleri oluşturul muştur. Bu modeller oluşturulurken ilke olarak; servis gör mek için sisteme gelen elemanların kuyrukta bekleme süre lerinin maliyeti ve bir servis vericinin boş zamanının sis teme olan maliyeti esas alınarak; ekonomik servis verici sayısı ve bir servis vericiye tahsis edilecek ekonomik eleman sayısı belirlenmiştir. Bölüm.1 de; konuyla ilgili olarak yapılan çalışma lardan bahsedilmiştir. Bölüm 2 de; sistem ve simülasyon kavramları ele alı narak tek kanallı, üç aşamalı bir kuyruk modeli simülasyon yoluyla incelenmiştir. Bölüm 3 de; kuyruk sistemleri ana hatları ve temel özellikleri ile ele alınmıştır. Bölüm 4 de; çeşitli kuyruk problemleri için maliyet modelleri oluşturulmuştur. Burada M/M/c ve M/M/c/K model leri için simülasyonla üretilen sayılar kullanılarak eko nomik kanal sayıları tesbit edilmiştir. (M/M/l) : (GD/M/M) modeli için de servis elemanının ücreti ve müşterilerin bek leme maliyeti esas alınarak bir servis vericiye (kanal) tahsis edilecek ekonomik eleman sayısı belirlenmiştir. Ayrıca bu model için iki ayrı pertürbasyon tekniği gelişti rilerek sonuçlar karşılaştırılmıştır. Yine bu bölümde M/M/c/K modeli için kayıp müşteri sayısının beklenen değerine ait bir formül oluşturarak, toplam maliyeti üzerinde kayıp müşterilerinin maliyetinin etkisinin olmadığı gösterilmiştir. Bölüm 5 de; Adapazarı Devlet Hastanesi 'nde M/M/c mo delini kullanarak ekonomik yatak sayısının nasıl değişmesi gerektiği belirlenmiştir. Ayrıca sistemi, varış ve servis bitim anlarında göz lemesi esasına dayanan G/M/l ve M/GD/1 modelleri ile ince leyip ekonomik yatak sayısını veren bir bağıntı oluşturul muştur.

the interval t 0,1]. After finding the random number R.which is uniformly distributed on the interval [0,1], one can find the value of the random variable x which depends on R by means of the formula x=F (R). One of the problems which arises here is the expression of F~ (R) in such a way that it is easy to work with it, so* to find the value of x is possible. Several. ways.of handling the problem and the generation of random variables for expo nential, gamma, normal, Poisson and binomial densities were expressed. In this chapter a queue formed by a service channel and a' service station which consists of three stages was investigated. The arrivals were considered as determinis tic and the service rates as exponantial. The random num ber which were used for 'service rates were produced..through simulation. For the values of the interarrivals X`^= 4, ^ =6, A--1- =8/ A~ =10 and for the values of the service rates m -,=20, /i2=25, m., = 35. The system was examined ? from the point of view of balancing the idle time of clerks and the waiting times of customers. Here y, is the service ? rate of the first stage of the. service system and so on. For such a service system the interarrival rates A~-'-=8, or A~1=10 is preferable depending on the -cost of the clerk idle time or the waiting time of the customers. In any case A~ -4,..or A`` -6 are not economical rates. In Chapter 3, the basic properties and the formula tion of the parameters of some queueing models were exami ned. Some instruments which are very useful in the moder-. ling of the structures of queueing problems were expressed. A queyeing system is completely determined by these six factors;-' 1. Input or arrival (interarrival) distribution. ' 2. Output or departure (service) distribution. 3. Service channels. 4. Service discipline. 5. Maximum number of customer alloved in the system. 6. Calling source. A notation developed by Kendal, Lee and Taha for multiple server queueing models and called after them as Kendal-Lee-Taha^ notation was given. With the help of' this notation... the-. characteristics of the model was expressed explicity..'These characteristic determines the type of. the model uniquely. The formulations of Chapman-Kolmogorov backward and forward equations and the infinitely small generator matrix Q of the birth-death process, a simple proof of Little for mula for Poisson input, general service first- in-first-out queue discipl'ine were given. State- transition-rate-diagram was designed for brith-death process. By making use of the same argument, state-transition -rate- diagrams were also xridesigned for (M/M/m) : (FİFO/°°/?°), (M/M/l) :FÎFO/K/°°), (M/M/m) : (FÎFO/M/00) /, (M/M/l ): (FİFO/K/K). With the help of these diagram difference equation are formed for each of them for the case of stationary state. ; In Chapter 4, the formulation of cost models for so me queueing models under various considerations were pre sented. A model which aimes the minimization of the cost arising from the idle time of the clerk.-and the waiting time of the. customers for the case where ''arrival'.and servi ce rates were obtained from the Poisson distribution which they belong, to by means of simulation is developed and for M/M/c and M/M/c/K the number of clerks which minimizes the cost function are found by means of the relation E(c) :C =C `c-r+RL (!) s q here I :idle time of a. clerk W. :the waiting time of a customer C.-marginal cost of a clerk for unit time c :number of parallel services A : arrival rate M- :.service rate E(C) :the cost arising from the idle time of clerks and the waiting times of customers For several values of the parameters involved in the model C' was plotted versus r/= A/a< in such a way that- the value of c which minimizes C' could be found by drawing a line parallel to C' from a given value -of r. Since' the values of X and m so r are. obtained by means. of simulation and the process is continuously in operation the system is observable. and under control at every moment.. If c is the minimum of E(c), then E(c)<E(c-l), E(c)<E(c+l) Combining these relations with (1) we obtain 1 1 «R < (2) L (c-l)-L (c) Lq(c)-Lq(c41) and the value of c which satisfies (2) will.'.be the number of parallel servers which minimizes' the cost model. The results obtained from (2) were compared with the results of (1) for severel values of the parameters involved in the models and were seen that for the same values of the parameters they give the same solutions for c. For (M/M/l) : (GD/m/m) model the economic number of customers assigned to server was found by considering the cost of servers idle time and the waiting time of the x`i iicustomers. In this context we used the relation,-, / n ` i 1-f RL(m) /o» E(m) : C = m1 =-~ - - (3) s in A graphical representation of the results obtained for several values of the parameters involved in the formula tion of the problems was given. Arrival rates and service rates are generated by simulation in such a manner that an observer could keep the system under. control and apply the measures at the moment they were needed. While We looked for the economical value of m two different per turbation techniques were used and their results were compared with results obtained from (3). For M/M/c/K model we constructed a formula for the expected value of the numbers of the lost customers by considering the cost arising due to the custemers which are lost since there were no available place to wait for service. Here we also used simulation and found several values of the cost changing due to the changes in K. For small values of K the portion of the cost which aroused due to lost custemers in total cost was quite clear but this partion decreased steadly while K increased. In Chapter 5, the data obtained from the Child Clinic of Adapazarx State Hospital were used to find the mean of days a child spent in the hospital to regain his (her) health' for each mount. We also found the service, rate of a bed and the arrival rates directed to the departmana. as a whole and to a single bed. We used M/M/c model to find the economic number. of beds for several, values of the cost arised due to the waiting time of a customer* and the empty time of a bed. By indicating the ratio of these two cost with R and changing R from 2 to 4 by an amount of 0.5 at each time it was shown that the economic number of bed should be in the interval (10,27). In order to find the economic number of beds a different approach was used by observing the system only at the times of arrivals or deperture points. If the system is observed at arrival times a birth to G/M/l system. If the departure points are in consideration then M/GD/1 will arise. For M/GD/1 the elements of the transition matrix of Markov chain are determined by the probabilities of number of arrivals during a service time and given by je-U(U)k b(t)dt qk o k i XXVFar G/M/l the elements of the transition matrix of Markov chain are determined by the probabilities of number ser vices during an interarrival time.,,k -nv d, - / -. - - a (v)dv K o k! the results obtained from the models formed by observing the system only at the times of arrivals or at the times of departures are a little bit different from the results obtained from the models formed by continujOiu.s^observation., The reason for this difference caused by the way of ob-.' servation. An advantage of observing the queueing system at only arrival times or at departure times is to give the opportunity to an analyst to reduce the system to a disc rete time Markov chain even though the orijinal system is non Markovian. xv