الفهرس 
THE QUEUE M/M/1 WITH ADDITIONAL SERVERS FOR A LONGER QUEUEOnly 14 pages are availabe for public view
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Abstract
All of us have experienced the annoyance of having to wait in line. Unfortunately, this phenomenon is becoming more and more prevalent in our increasingly congested and urbanized society. We wait in line in our cars in traffic jams or at toll booths; we wait in line at supermarkets to check out; we wait in line at post offices; and so on, ad infinitum. We, as customers, do not generally like these waits, and the managers of the establishments at which we wait also do not like us to wait, since it may cost them business. Why then is there waiting? the answer is relatively simple. There is more demand for service than there is facility for service available. Why is this so? there may be many reasons; for example, there may be a shortage of available servers; it may be infeasible economically for a business to provide the level of service necessary to prevent waiting; or there may be a space limit to the amount of service that can be provided. Generally, these limitations can be removed with the expenditure of capital, and to know how much service should then be made available, one would need to know answers to such questions as, “How long must a customer wait?” and “How many people will form in the line?” Queuing theory attempts (and in many cases, succeeds) to answer these questions through detailed mathematical analysis.
The objective of this thesis is to analyze some queuing problems to obtain suitable forms for characteristics of each queuing servicing system.
The thesis consists of four chapters.
The first chapter deals with the queuing system M/M/1 with additional servers for a longer queue. Suppose that the customers arrive to the system individually in a Poisson process and service individually with negative exponential service times. The service discipline is FCFS. As long as the number of customers in the system is greater than or equal to zero and less than or equal to N, there is only one server in the system. As the number of customers in the system increases to more than N and is still less than or equal to 2N, an additional server is added. This additional server is removed when the number of customers in the system decreases to N or less. As soon as the number of customers in the system goes beyond 2N, the number of servers will be three. Similarly, the third server will be taken off when the number of customers falls to 2N or below. Clearly the traffic intensity for this system will depend on the number of additional servers. The expected number of customers in the system, the probability of the additional of one server and the probability of the additional of two servers are obtained under the assumption that the number of additional servers depends on the number of customers in the system. The condition under which the M/M/1 queuing system with additional servers is profitable is discussed. A MATLAB program is used to illustrate this condition numerically. Finally, the maximum likelihood estimators for the parameters of this queuing system are obtained. The results obtained in this chapter are considered as a special case from the results that obtained by Mokaddis and Zaki [48].
The material of this chapter has been published in International Journal of Modern Engineering (IJMER), [50].
The second chapter concerns the queuing system M^X/G^(1,B)/1, to which the customers are assumed to arrive in batches of random size X according to a compound Poisson process and also are served in batches. As soon as the system becomes empty, the server leaves for a vacation of random length V. If no customers are available for service after returning from that vacation, the server keeps on taking vacations till he finds at least one customer in the queue, then immediately begins to serve the customers up to the service capacity B. If more than B customers are present when the server returns from a vacation, the first B customers are taken into service. If fewer than B customers are present, all waiting customers go into service. Late arrivals are not allowed to join the ongoing service even if there is space available. The steady state behavior of this queuing system is derived by an analytic approach to study the queue size distribution at a random point as well as at a departure point of time under multiple vacation policy. It may be noted that the results in Gautam Choudhury [22] can be obtained as special cases from the results in this chapter when letting B=1 in this chapter, also the results in Lee et. all, [38] can be obtained as special cases from the results in this chapter when letting X=1 and the queue with single vacation in this chapter.
The material of this chapter has been published in International Journal of Modern Engineering (IJMER), [51].
The third chapter undertake the transient analysis of a limited capacity queuing system with three operating states in presence of catastrophes. The customers arrive at the system one by one in accordance with a Poisson process at a single service station. The customers are served one by one at the single channel. The service time is exponentially distributed, and the system with three operating states E, F, H. When the system is not empty, catastrophes occur per a Poisson process with rate ξ. The effect of each catastrophe is to make the queue instantly empty. Simultaneously, the system becomes ready to accept the new customers. The queue discipline is first-come-first-served and the capacity of the system is limited to N. The direct application of the model can be ascribed to access of internet to a computer by three operating states [USB modem, DSL internet (Telephone Line) and Wi-Fi hotspot (Mobile Internet)]. The catastrophes may occur with each of these three states and make the number of programs work with internet zero instantaneously. Then the number of these programs can be estimated by using the described queuing model with operating change and catastrophes. The effects of operating change and catastrophes are extensively dealt with. The steady state behavior of the queuing system is also discussed and the mean queue length is obtained. Finally, a particular case is considered to yield well-known results of the M/M/1 queue with finite and with infinite waiting space, and also we can obtain some corrected results for the corresponding results obtained by Jain and Kanethia [29]. It may be noted that the results in Jain and Kanethia [29] can be obtained as special cases from the results in this chapter if we consider only two operating states.
The fourth chapter treats with a recursive method to the optimal control of an M/E_K/1 queuing system with a removable service station under steady-state conditions and N-policy, i.e., the server is turned on when the number of customers in the system reaches a certain number, N(N≥1), and is turned off when there are no customers in the system. It is assumed that customers arrive following a Poisson process with parameter λ and with service times per an Erlang distribution with mean 1/μ and stage parameter k. The Erlang type k distribution is made up of k independent and identical exponential stages, each with mean 1/kμ. A customer goes into the first stage of the service (say stage k), then progresses through the remaining stages and must complete the last stage (say stage 1) before the next customer enters the first stage. We assume that customers arriving at the service station form a single waiting line and are served in the order of their arrivals; that is, the first-come first-served discipline. It is further assumed that the station can serve only one customer at a time, and that the service is independent of the arrival of the customers. If the service station is busy, then a customer must wait until the station is available. Analytic closed-form solutions of the controllable M/E_K/1 queuing system are derived recursively and also by using p.g.f technique. This is a generalization of the controllable M/M/1, the ordinary M/E_K/1, and the ordinary M/M/1 queuing systems in the literature. The expected number of customers in the system, which is equivalent to the sum of the expected number of customers in the system when the service station is turned off and the expected number of customers in the system when the service station is turned on, is computed by using probability generating function techniques. We also prove that the probability that the service station is busy in the steady-state is equal to the traffic intensity. Following the construction of the expected cost function per unit time, we determine the optimal operating N-policy at which the cost function is minimum.