M/G/1/K與G/M/1/K排隊含啟動時間的F方策與N方策之相互關係

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國

立

交

通

大

學

工業工程與管理學系

博

士

論

文

M/G/1/K與G/M/1/K排隊含啟動時間的

F方策與N方策之相互關係

Interrelationships between F Policy and N Policy for

M/G/1/K and G/M/1/K Queues with Startup Time

研究生：郭清章

指導教授：彭文理教授

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M/G/1/K與G/M/1/K排隊含啟動時間的

F方策與N方策之相互關係

Interrelationships between F Policy and N Policy for

M/G/1/K and G/M/1/K Queues with Startup Time

研究生：郭清章 Student：Ching-Chang Kuo

指導教授：彭文理博士 Advisor：Dr. W. L. Pearn

國立交通大學

工業工程與管理學系

博士論文

A Dissertation Submitted to

Department of Industrial Engineering and Management College of Management

National Chiao Tung University in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in

Industrial Engineering and Management

November 2007

Hsinchu, Taiwan, Republic of China

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Acknowledgment

誌謝

很高興能夠在三年多的時間內完成本論文，在撰寫論文期間多虧許多人教導與協助，使我在困境中能找到方向，也因此讓我感受到師友的可貴。在此，首先要感謝的是指導老師彭文理教授，在老師細心教導下，學生可以學到更多知識，再者是撰寫論文過程中，得到中興大學王國雄教授的從旁協助，讓學生可以加速完成論文，此外還要感謝本系鍾淑馨教授、中興大學應數系王國雄教授、臺灣科技大學工管系徐世輝教授及臺中技術學院統計系柯沛程教授，對本論文費心的審查及提供諸多的寶貴意見使本論文更加完備。特別感謝是雅甄同學，在博士修業期間不斷鼓勵與關懷，使我能夠保持愉悅的心情學習，再者，要謝謝交大實驗室裡其他學長姐、學弟及同學們在課業上的砥礪與啟發，使我獲益良多。最後要謝謝家人支持與關心，尤其是二哥郭清寶，在學習過程中不斷照顧與支持，讓我能夠一路順利完成學業，完成我的夢想。謹以此獻給所有關愛我的家人。郭清章謹誌於國立交通大學管理學院工業工程與管理學系中華民國九十六年十一月二十一

日

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M/G/1/K 與 G/M/1/K 排隊含啟動時間的

F 方策與 N 方策之相互關係

學生：郭清章指導教授：彭文理博士

國立交通大學管理學院

工業工程與管理學系

摘要

此篇論文是在研究F 方策 M/G/1/K 和 G/M/1/K 排隊含起動時間的問題，進而 深入探討 F 方策和 N 方策的相互關係。在排隊問題中，F 方策主要是研究控制到達的 問題。N 方策排隊問題主要是研究控制服務的問題。首先，我們探討對於在 F 方策 M/G/1/K 和 G/M/1/K 排隊含起動時間的問題。而 F 方策的定義如下：當顧客數目 到達系統可承載數(例如系統容量)，系統不再允許任何到達顧客進入，直到有一定數目 (足夠)的顧客已被服務，也就是顧客數目會降到一個門檻值 F (0≤ <F K −1)。同時，啟動系統開始讓顧客進入系統，其啟動時間的分配為指數分配，參數為β 。因此，系統會正常運作直到系統裡的顧客數目到達系統可承載的數目，所有的程序會再重新依序發生。我們分別針對在F 方策 M/G/1/K 和 G/M/1/K 排隊中提出遞迴的方法與輔 助變數技巧來推導。其方式如下，使用輔助變數代替剩餘服務時間(到達時間)再利用遞迴方法來計算在穩態下的機率。為了分析說明遞迴方法，此篇論文提出三種不同服務時間(到達時間)分配來解釋在 F 方策 M/G/1/K 和 G/M/1/K 排隊系統含指數分配的起動時間，其服務時間(到達時間)分配包括指數分配、3 階段 Erlang 和 deterministic 分配等。同時，針對最佳化的問題，建立其成本模型來決定最佳F 值，使得成本最小。 我們也使用Maple 電腦程式來計算出最佳 F 值與系統參數的關係，並作敏感度的分析。 為了進一步探討F 方策和 N 方策的關係，我們同樣使用遞迴方法和輔助變數技巧來求 取含啟動時間之 N 方策 M/G/1/K 排隊系統中穩態機率的演算法。而 N 方策的定義 如下：當顧客數目增加到一個門檻值N (N ≥ 1)時，啟動系統開始服務，其啟動時間的分配為指數分配，參數為γ 。且直到系統內沒有顧客後關閉服務。透過一系列的演算法的比對與計算，驗證出F 方策和 N 方策之間的互補關係：可由 F(或 N)方策排隊系統 所得到的演算法去計算另一個方策的排隊問題的解。最後，我們提供兩個範例，分別為3 階段 Erlang 與指數分配來作說明 F 方策和 N 方策之間的互補關係。 關鍵字：F 方策; N 方策; M/G/1/K 排隊; G/M/1/K 排隊; 遞歸方法; 啟動時間; 輔助變數; 成本; 敏感度分析.

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Interrelationships between F Policy and N Policy for

M/G/1/K and G/M/1/K Queues with Startup Time

Student: Ching-Chang Kuo Advisor: Dr. W. L. Pearn

Department of Industrial Engineering and Management National Chiao Tung University

Abstract

This dissertation deals with the interrelationship between F policy and N policy. The F policy queuing problem investigates the most common issue of controlling arrival to a queuing system. The N policy queuing problem investigates the most common issue of controlling service. The optimal control arrival in M/G/1/K and G/M/1/K queues operating under the F policy and startup time is investigated in this dissertation. The definition of F policy is described as following: When the number of customers in the system reaches its capacity K (i.e. the system becomes full), no further arriving customers are allowed to enter the system until there are enough customers who have been served in the system. Consequently, the number of customers in the system decreases to a threshold value F (0≤F <K−1). At that time, the server requires to take an exponential startup time with parameter β to start allowing customers in the system. Thus, the system operates normally until the number of customers in the system reaches its capacity at which time the above process is repeated all over again. A recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining service (or inter-arrival) time, is provided to develop the steady-state probability distributions of the number of customers in two finite queues. To illustrate analytically the two recursive methods, examples of different service (or interarrival) time distributions, such as exponential, 3-stage Erlang and deterministic distributions, in the F policy M/G/1/K queuing system and in the F policy G/M/1/K queuing system with exponential startup time distribution is present. In both queueing systems, a cost model is established to determine the optimal management F policy at minimum cost. An efficient Maple computer program is used to determine the optimal operating F policy and some system performance measures. Sensitivity analysis is also studied. To find the interrelationship between F policy and N

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policy, we have solved the solution algorithm of the N policy M/G/1/K queue with startup time. A recursive method and supplementary variable technique to obtain the solution algorithm is provided. The definition of N policy is described as following: The server needs a startup time when the number of customers in the system reaches the threshold N N ≥ for the first time until there are no customers present in the system. ( 1) At that time, the server needs to take an exponential startup time with parameter γ to start servicing customers in the system. Through a series of the algorithm, the complementary interrelationship between the F policy and N policy queues is obtained. Therefore, the problem of F policy (N policy) queuing system with startup time gives the solution algorithm to the other problem. The two simple examples of 3-stage Erlang and exponential distribution to illustrate the interrelationship are provided.

Keywords: F policy, N policy, G/M/1/K queue, M/G/1/K queue, Recursive method, Startup times, Supplementary variable, Cost, Sensitivity analysis.

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List of Contents

Page

Acknowledgment···i

Abstract (Chinese)···ii

Abstract (English) ···iii

List of Contents ···v

List of Tables ···vii

Chapter 1. Introduction···1

1.1 Background··· 1

1.2 Theoretical Analysis Techniques··· 3

1.3 Literature Review···5

1.4 Problem Statement ··· 6

1.5 Scope of Dissertation··· 8

Chapter 2. The F Policy M/G/1/K Queue with Startup Time ···9

2.1 Assumptions and Notations··· 9

2.2 Development of the Equations and Solutions ··· 11

2.3 Steady State Results···12

2.3.1 Recursive methods···13

2.3.2 The solution algorithm···17

2.4 Simple Examples···18

2.5 Optimal F policy···24

2.5.1 Cost function···24

2.5.2 Numerical examples···25

Chapter 3. The F Policy G/M/1/K Queue with Startup Time ···27

3.1 Assumptions and Notations···27

3.2 Development of the Equations and Solutions···28

3.4 Simple Examples···34

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3.5.1 Cost function···39

3.5.2 Numerical examples···40

Chapter 4. The N policy M/G/1/K Queue with Startup Time ···43

4.1 Assumptions and Notations···43

4.2 Development of the Equations and Solutions···44

4.4 Simple Example···50

Chapter 5. Interrelationship between the F policy and the N policy for M/G/1/K and G/M/1/K Queues with Startup Time ···53

5.1 Development of the F policy G/M/1/K Queue···53

5.2 Interrelationship between the N policy M/G/1/K Queue and the F policy G/M/1/K Queue···55

5.3 Development of the N policy G/M/1/K Queue···57

5.4 Interrelationship between the F policy M/G/1/K Queue and the N policy G/M/1/K Queue···60

Chapter 6. Conclusions and Future Researches···63

6.1 Conclusions···63

6.2 .Future Researches···64

Appendix···66

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List of Tables

Page

Table 1 The optimal value of F and its minimum expected cost for the service time distribution such as exponential ···26

Table 2 The optimal value of F and its minimum expected cost for the service time distribution such as 3-stage Erlang··· 26 Table 3 The optimal value of F and its minimum expected cost for the service time

distribution such as deterministic···26

Table 4 The optimal value of F and its minimum expected cost for exponential interarrival time ···42

Table 5 The optimal value of F and its minimum expected cost for 3-stage Erlang interarrival time ···42

Table 6 The optimal value of F and its minimum expected cost for deterministic interarrival time ···42

Table 7 N policy M/G/1/K queue corresponds to F policy G/M/1/K queue ···55

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Chapter 1 Introduction

In section 1.1, we describe the background of queueing theory that has been a continuously growing and interesting science. In section 1.2, an important technique that is supplementary variable technique will be introduced. In section 1.3, several researchers are shown in earlier works. In section 1.4, the description of the problems that are the interrelationships between F policy and N policy in M/G/1/K and G/M/1/K queues are presented. In section 1.5, we illustrate the scope of dissertation.

1.1 Background

The most of the optimization problems in queueing systems focused on design or static models in which the system characteristics did not vary with time in the last few years. It is evidence that this type of model does not meet the requirements of the majority of realworld queueing applications. For example, those models relative to the management of large-scale systems in several fields are: distribution, transportation, administration, production, informatics, etc. It is especially true in many computer and communication applications, in which the performance of the investigation system may be improved if some system parameters are adjusted as the system state changes. (see Kleinrock [24], Bolch et al. [6])

Hence, it is more concerned that the system characteristics are allowed to vary with time in the control or dynamic models. The aim of such models is to prescribe a certain behavior on the part of the decision maker. In these models, it is emphasized that the optimization is achieved over a class of operating policies, rather than over a set of parameters for a single operating policy, which is a fixed part of the model. A complete survey about the design and control of models was presented by Gross and Harris [15], and Crabill et al. [9].

Several researchers investigate five kinds of controllable queues as follows:

1. The N policy was first developed by Yadin and Naor [36] in 1963. When the number of customers in the system reaches the threshold N(N ≥1) for the first time until there are no customers present in the system, the server returns to provide

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service.

Many researchers have worked on this subject such as Yadin and Naor [36], Hersh and Brosh [16], Kimura [23], Lee and Park [26], Medhi and Templeton [28], Takagi [29], Wang et al. [32], Wang and Ke [33], Ke and Wang [20] and others.

2. The F policy was developed by Gupta [12] in 1963. When the number of customers in the system reaches its capacity K (i.e. the system becomes full), no further arriving customers are allowed to enter the system until there are enough customers in the system have been served so that the number of customers in the system decreases to a threshold value F (0≤ <F K−1). At that time, the server requires to take an exponential startup time to start allowing customers in the system. Thus, the system operates normally until the number of customers in the system reaches its capacity at which time the above process is repeated all over again.

3. The T policy was developed by Heyman [18], Levy and Yechiali [27], Tijm [31], Gakis et al. [11], and Ke[22]. Following the beginning of the idle period, the server returns to provide service immediately after T time units have elapsed from the epoch of server removal if there is at least one customer present in the waiting line, until there are no customers in the system. If after T time units have elapsed, there are no customers in the system to initiate service, the server waits another T time units, and so on, until at least one customer is present.

4. The D policy was developed by Balachandran [3], Tijm [31] and Gakis et al. [11]. The server returns to provide service if the accumulated backlog, which is the sum of service time of the new arriving customers, exceeds a given quantity

D(D≥ 1) for the first time.

5. The combined policies which are combined by any two of the N, T, and D policies. Doganata [10] first studied the NT policy M/G/1 queueing system that the vacation period is terminated if the time elapsed since the first arrival during the vacation period reaches the threshold T, or the number of customers in the system waiting for services reaches the threshod N. Alfa and Frigui [1] extended Doganata's model to the MAP/PH/1 case. Gakis, et al. [11] presented six dyadic policies for the M/G/1 queueing system.

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1.2 Theoretical Analysis Techniques

Recall that a queue is characterized by the input process, the service mechanism, and the queue discipline. When arrivals have Poisson characteristics and service times are exponential, the resulting queueing process is Markovian. A Markovian model is in the framework of the birth-and-death process that completely specifies the state of the system at a given time. This information is sufficient to describe the future development of the process. These assumptions imply that the future evolution of the system from some time t depends only on the state of the system at time t , and is independent of the history of the system prior to time t . In these models, the "state" of the system could always be specified in terms of the number of customers present. (In a multidimensional case, the state is specified in terms of the number of customers of each type present at time t .)

Suppose that we are interested in a queue for which the number of customers present at any time t is not sufficient information to permit complete analysis of the model. Such a queue is impossibly solved by using birth-and-death process. Clearly, some methods are required.

In this section, we will introduce an important technique to study the F policy M/G/1/K and G/M/1/K queues with startup time and N policy M/G/1/K queue with startup time. It is supplementary variable technique, introduced by Cox [8].

Suppose that customers arrive at random with rate α in a single server queue. Let the service times of customers be independently distributed with p.d.f. b t ( ) and hazard function ( )h t . When ( )_b b t is not an exponential distribution, the

probability of service being completed in ( ,t t+ Δt depends on the length of time ) service has been in progress.

If the customer currently being served has been at the service point for a time

u. Let p u t be the joint probability density of the state ( , )_n( , ) n u at time t . More

explicitly, if U is the random variable corresponding to u, + Δ → < < + Δ = Δ 0

Pr( and customers present at time )

( , ) lim . n u u U u u n t p u t u

Then we can have the forward equations for the process using the usual argument as in the birth and death model.

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The following forward difference equations may be obtained in steady-state, for 1,2,...n = 0( ) 0( )(1 ) ₀ 1( ; ) ( )b ( ), p t dt+ = p t −αdt +

∫

∞p u t h u du+ Δ dt (1.1)

{

}

1 ( , ) ( , ) 1 [ ( )] ( , ) ( ), n n b n p u dt t dt+ + = p u t − α+h u dt +p ₋ u t dtα + Δ dt (1.2) and 1 0 1, 0 (0, ) ( ; ) ( ) ( ) ( ), n n b n p t =

∫

∞p + u t h u du+αp t δ + Δ dt (1.3)

where δ_1,_j =1 if j =1 and δ_1,_j =0 otherwise.(δ_{1, j} is the Kronecker delta function.) By performing the following expansions to (1.2)

( , ) ( , ) ( , ) ( , ) n n ( ), n n p u t p u t p u dt t dt p u t dt dt dt u t ∂ ∂ + + = + + + Δ ∂ ∂

we obtain the following partial differential equations

1 ( , ) [ ( )] ( , ) ( , ). n b n n p u t h u p u t p u t u t α α − ∂ ∂ ⎛ ₊ ⎞ _{= − +} ₋ ⎜_∂ _∂ ⎟ ⎝ ⎠ (1.4)

Solutions to (1.1) and (1.3)-(1.4) can be obtained with the help of mathematical theory and techniques.

Let u denote the length of time when customer is being served at time t until his service completion. We have the following backward difference equations may be obtained in steady-state, for n =1,2,....

0( ) 0( )(1 ) 1(0, ) ( ), p t dt+ = p t −αdt +p t dt+ Δ dt (1.5) α ₋ α ₊ − + = − + ₁ + ₁ + Δ ( , ) ( , )(1 ) ( , ) (0, ) ( ) ( ). n n n n p u dt t dt p u t dt p u t dt p t b u dt dt (1.6)

By performing the following expansions to (1.6)

( , ) ( , ) ( , ) ( , ) n n ( ), n n p u t p u t p u dt t dt p u t dt dt dt u t ∂ ∂ − + = − + + Δ ∂ ∂

the following partial differential equations are obtained

1 1 ( , ) ( , ) ( , ) (0, ) ( ). n n n n p u t p u t p u t p t b u u t α α − + ∂ ∂ ⎛ ₊ ⎞ _{= −} ₊ ₊ ⎜_∂ _∂ ⎟ ⎝ ⎠ (1.7)

Solutions to (1.1)-(1.7) can be solved with the help of mathematical theory and techniques.

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1.3 Literature Review

We use a supplementary variable technique to analyze the optimal control of the F policy M/G/1/K and F policy G/M/1/K queues where the server needs a startup time before start allowing customers in the system and K < ∞ denotes the maximum capacity of the system. The method of controlling arrivals focuses on reducing the number of customers in the system. The model proposed in this dissertation is very useful in real-life situations since the controlling of arriving customers is considered.

Steady-state analytical solutions of the F policy M/M/1/K queueing system with an exponential startup time were first developed by Gupta [12]. However, steady-state analytical solutions of the F policy queue with interarrival times or service times distribution of the general type have not been found. It is extremely difficult, if not possible, to obtain the explicit expressions for the steady-state probability distribution of the number of customers in the system. This becomes particularly helpful when the supplementary variable technique to the non-Markovian queueing system having general interarrival times or general service times is used. Cox [8] first introduced the supplementary variable technique. Based on this technique, Gupta and Rao [13-14] presented a recursive method to develop the steady-state probability distributions of the number of failed machines for the no-spare M/G/1 machine repair problem and the cold-standby M/G/1 machine repair problem, respectively.

Past work regarding queues may be divided into two parts according to whether the system is considered to control the service or the arrival. In the first category of controlling the service, the N policy M/M/1 queueing system without startup was first introduced by Yadin and Naor [36]. The extension of this model can be referred to Bell [4-5], Heyman [17], Kimura [23], Teghem [30], Wang and Ke [33], and others. Wang and Ke [33] provided a recursive method and used the supplementary variable technique to develop the steady-state probability distributions of the number of customers for the N policy M/G/1/L queueing system. Ke and Wang [20] presented a recursive method and applied the supplementary variable technique to obtain the steady-state probability distributions of the number of customers for the N policy G/M/1/L queueing system. The server startup corresponds to the preparatory work of the server before starting the service.

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In some real-life situations, the server often needs a startup time before beginning to provide the service. Several authors research on queueing systems with startup time focus mainly on the N policy M/G/1 queues. Baker [2] first studied the N policy M/M/1 queueing system with an exponential startup time. Borthakur et al. [7] extended Baker’s model to the general startup time. The N policy M/G/1 queueing system with startup time was investigated by several researchers such as Medhi and Templeton [28], Takagi [29], Lee and Park [26], Hur and Paik [19], Krishna et al. [25], Ke [21], Wang and Ke [33],Wang and Ke [34], Wang et al.[35], and so on. Ke [21] presented a recursive method and used the supplementary variable technique to compute the operating characteristics for the N policy G/M/1/L queueing system with an exponential startup time. In the second category of controlling the arrivals, the analytical developments for controlling the arrivals in queueing problems are rarely found in the literature, which are particularly for service time and interarrival time following general type. The work of related problems in the past mainly concentrates on Markovian system. The pioneering work in steady-state analytical solutions of the F policy M/M/1/K queueing system with an exponential startup time was first derived by Gupta [12]. Through a series of propositions, the interrelationship between the operating N policy and the operating F policy are established by Gupta [12].

Practically, the memoryless property of the arrival (input) process does not always meets the needs of applications because, for interarrival time, general distribution, rather than exponential distribution, appears to be more appropriate and reasonable. General distribution can include the special cases of exponential, Erlang, hyper-exponential, and deterministic, etc. However, aside from theoretical arguments, many real-life situations satisfy the assumptions of Markovian conditions for service time. Hence, we may consider inevitably to analyze the F policy M/G/1/K and F policy G/M/1/K queues.

1.4 Problem Statement

In this dissertation, we investigate the interrelationship between F policy and N policy in M/G/1/K and G/M/1/K queues. First, we study the optimal control in the F policy M/G/1/K and G/M/1/K queues. The definition of a F policy is described as follows: When the number of customers in the system reaches its

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capacity K (i.e. the system becomes full), no further arriving customers are allowed to enter the system until there are enough customers in the system have been served so that the number of customers in the system decreases to a threshold value F (0≤ <F K −1). At that time, the server requires to take an exponential startup time to start allowing customers in the system. Thus, the system operates normally until the number of customers in the system reaches its capacity at which time the above process is repeated all over again. Gupta [12] first developed the concept of an F policy steady-state analytical solution of the F policy M/M/1/K queueing system with an exponential startup time.

A number of practical problems arise which may be formulated as one in which the server requires taking a startup time to start allowing customers in the system. Such models have potentially useful in practical real-life. For example, in computer process and service systems, massages are transmitted among the computers (processors). If the processor is free the message is accepted; otherwise the message is temporarily stored in a buffer to be served some time later. When the buffer is full, the arriving messages will be restricted entrance until the number of messages drops to some a threshold level. When system buffer reduces to the threshold level, the messages are immediately admitted to enter the system. This will help to prevent the system from becoming over-loaded. Another application of our model is transportation. In order to avoid traffic jams caused by motorists returning home for Chinese New Year, the entrance ramps along the highway will be controlled by a metering system. When traffic flow is congested, entrance ramps are closed to keep expressway traffic smooth. Vehicles are allowed to re-enter once the traffic is improved. The entrance ramps may need to maintain and the service may be temporarily shut down. The model is also applicable to controlling the amount of eco-tour visitors, e.g. in Kenting National Park (Taiwan). When applicants reach the limited numbers of the day, the application would be rejected.

We will study the interrelationship between F policy and N policy in M/G/1 and G/M/1 queues. We first consider the N policy M/G/1 queue with startup time. The decision-maker can turn a single server on at any arrival epoch or off at any service completion (departure) epoch. The term 'removable server' is just an abbreviation for the system of turning on and turning off the server, depending on the number of customers in the system. Yadin and Naor [36] first introduced the

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concept of an N policy which turns the server on when N N ≥( 1) or more customers are present and turns the server off only when the system is empty. After the server is turned off, the server may not operate until N customers are present in the system.

Suppose that the time elapsing between two successive arrivals is independently and identically distributed (i.i.d.) random variable, having a general distribution A v v ≥( ) ( 0), a probability density function (p.d.f.) a v v ≥( ) ( 0) and mean interarrival time a . The service times of successive customers are ₁

independent and identically random variables having a common distribution ( ) ( 0)

S u u ≥ , a probability density function s u u ≥( ) ( 0) and mean service time s . ₁

The service process is independent of the arrival process. We assume that arriving customers form a single waiting line based on the order of their arrival; that is, the first-come, first-served discipline. The server can serve only one customer at a time. A customer, upon entry into the service facility, finding that the server is busy have to wait in the queue until the server is free.

1.5 Scope of Dissertation

In chapter 2, we provide a recursive method using the supplementary variable technique to derive the steady-state probability distributions in the F policy M/G/1/K queue. We illustrate the solution algorithm by presenting three simple examples for three different service time distributions: exponential (denoted M), 3-stage Erlang (denoted E3), and deterministic (denoted D). Various system performance measures are also presented. The total expected cost function per unit time is developed. Numerical and comparative results are shown. In chapter 3, we follow the above method of the chapter 2 and treating the supplementary variable as the remaining interarrival time to develop the F policy G/M/1/K queue with startup time. In chapter 4, we study N policy M/G/1/K queue with startup time. In chapter 5, we examine the relationships between the F policy and N policy. Using the solution algorithm of N policy (F policy) M/G/1/K queue with startup time, we develop the steady state probabilities of F policy (N policy) G/M/1/K queue. Finally, chapter 6 consists of some concluding remarks.

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Chapter 2 The F Policy M/G/1/K Queue with Startup Time

A supplementary variable technique is used to study the optimal management problem of the F policy M/G/1/K queue where the server needs a startup time before start allowing customers in the system and K < ∞ denotes the maximum number of customers in the system. The method of controlling arrivals focuses on reducing the number of customers in the system. The model presented in this dissertation is very useful in real-life situations since the controlling of arriving customers is considered.

The primary objective of this chapter is threefold. Firstly, we develop a recursive method using the supplementary variable technique and treating the supplementary variable as the remaining service time, to develop the steady-state probability distributions of the number of customers for the F policy M/G/1/K queue. The method can be utilized for any service time distribution, such as deterministic (denoted D), exponential (denoted M) and k-stage Erlang (denoted Ek), etc. Secondly, to illustrate a recursive method we present three simple examples for three different service time distributions such as exponential, 3-stage Erlang, and deterministic. Thirdly, we study various system performance measures, such as the average number of customers in the system, the probability that the server is busy, the blocking probability, etc. The total expected cost function per unit time for the

F policy M/G/1/K queue with startup times is developed. Numerical and

comparative results are also provided. 2.1 Assumptions and Notations

We consider the controlling arrivals to a finite capacity M/G/1 queue with combined F policy and exponential startup time. It is assumed that customers arrive according to a Poisson process with parameter λ , and the service times of the successive customers are independently and identically distributed (i.i.d.) random variables having a distribution S u

( )

(

u ≥0

)

, a probability density function s u

( )

(

u ≥0

)

and mean service time s . The arrival process is independent of the service ₁

process. We assume that arriving customers form a single waiting line based on the order of their arrivals; that is, the first-come, first-served discipline. Suppose that the

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server can serve only one customer at a time. Customers entering into the service facility and finding that the server is busy have to wait in the queue until the server is available. Gupta [6] first introduced the concept of a F policy. The definition of a

F policy is described as follows: When the number of customers in the system

reaches its capacity K (i.e. the system becomes full), no further arriving customers are allowed to enter the system until there are enough customers in the system have been served so that the number of customers in the system decreases to a threshold value F (0≤ <F K−1). At that time, the server requires to take an exponential startup time with parameter β to start allowing customers in the system. Thus, the system operates normally until the number of customers in the system reaches its capacity at which time the above process is repeated all over again.

The following notations and probabilities are used throughout this chapter.

F threshold level

K system capacity

(

K > +F 1

)

S service time random variable

U remaining service time random variable

( )

S u distribution function (d.f.) of S

( )

s u probability density function (p.d.f.) of S

( )

*

S θ Laplace-Stieltjes transform (LST) of S

( )

_{( )}

* l

S θ l th order derivative of S*

( )

θ with respect to θ

( )

0,0

P t probability of no customers in the system at time t when the

arrivals are not allowed to enter the system

( )

0,n

P t probability of n customers in the system at time t when the

arrivals are not allowed to enter the system, where n=1,2,..., .K

( )

1,0

P t probability of no customers in the system at time t when the

arrivals are allowed to enter the system

( )

1,n

P t probability of n customers in the system at time t when the

arrivals are allowed to enter the system, where n=1,2,...,K − 1.

0,0

P steady state probability of no customers in the system when the arrivals are not allowed to enter the system

0,n

P steady state probability of n customers in the system when the arrivals are not allowed to enter the system, where n=1,2,..., .K

1,0

P steady state probability of no customers in the system when the arrivals are allowed to enter the system

1,n

P steady state probability of n customers in the system when the arrivals are allowed to enter the system, where n=1,2,...,K − 1.

1

s mean service time

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2.2 Development of the Equations and Solutions

We use the following supplementary variable: U ≡remaining service time for the customer in service. The state of the system at time t is given by

( )

N t ≡number of customers in the system, and

( )

U t ≡remaining service time for the customer being served. Let us define

( )

=

{

( )

= <

( )

≤ +

}

≥ = 0,n , Pr , d , 0, 0,1,..., . P u t du N t n u U t u u u n K

( )

=

{

( )

= <

( )

≤ +

}

≥ = − 1,n , Pr , d , 0, 0,1,..., 1. P u t du N t n u U t u u u n K

( )

=

_∫

∞

( )

= 0,n ₀ 0,n , d , 0,1,..., . P t P u t u n K

( )

=

_∫

∞

( )

= − 1,n ₀ 1,n , d , 0,1,..., 1. P t P u t u n K

Relating the state of the system at time t and t+dt , we obtain

( )

0,0 0,0 0,1 d 0, , dt P t = −βP t +P t (2.1)

( )

β

( )

₊

( ) ( )

∂ ∂ ⎛ ₋ ⎞ _{= −} ₊ _, _{≤ ≤} ⎜_∂ _∂ ⎟ ⎝ t u⎠P0,n u t, P0,n u t, P0, 1n 0,t s u 1 n F, (2.2)

( )

+

( ) ( )

∂ ∂ ⎛ ₋ ⎞ ₌ _, _{+ ≤ ≤} ₋ ⎜_∂ _∂ ⎟ ⎝ t u⎠P0,n u t, P0, 1n 0,t s u F 1 n K 1, (2.3)

( )

0,K , 1,K 1 , P u t P u t t u λ − ∂ ∂ ⎛ ₋ ⎞ ₌ _, ⎜_∂ _∂ ⎟ ⎝ ⎠ (2.4)

( )

1,0 1,0 0,0 1,1 d _{0, ,} dt P t = −λP t +βP t +P t (2.5)

( )

( ) ( )

λ β λ ∂ ∂ ⎛ ₋ ⎞ _{= −} ₊ ₊ ₊ ⎜_∂ _∂ ⎟ ⎝ ⎠ , 1,1 1,1 0,1 1,0 1,2 , , , 0, P u t P u t P u t P t s u t u P t s u (2.6)

( )

( ) ( )

λ β λ ₋ + ∂ ∂ ⎛ ₋ ⎞ _{= −} ₊ ₊ ⎜_∂ _∂ ⎟ ⎝ ⎠ + , ≤ ≤ 1, 1, 0, 1, 1 1, 1 , , , , 0, 2 , n n n n n P u t P u t P u t P u t s u t u P t s u n F (2.7)

( )

( ) ( )

1, , 1, , 1, 1 , 1, 1 0, 1 2, n n n n P u t P u t P u t s u P t s u t u F n K λ λ ₋ ₊ ∂ ∂ ⎛ ₋ ⎞ _{= −} ₊ ₊ _, ⎜_∂ _∂ ⎟ ⎝ ⎠ + ≤ ≤ − (2.8)

( )

1,K 1 , 1,K 1 , 1,K 2 , . P u t P u t P u t t u − λ − λ − ∂ ∂ ⎛ ₋ ⎞ _{= −} ₊ ⎜_∂ _∂ ⎟ ⎝ ⎠ (2.9)

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2.3 Steady State Results In steady state, let us define

( )

→∞ = = 0,n lim 0,n , 0,1,..., . t P P t n K

( )

→∞ = = − 1,n lim 0,n , 0,1,..., 1. t P P t n K

( )

→∞ = = 0,n lim 0,n , , 1,2,..., . t P u P u t n F

( )

→∞ = = − 1,n lim 1,n , , 0,1,..., 1. t P u P u t n K

and further define

( )

=

( )

=

0,n 0,n , 1,2,...,

P u P s u n F . (2.10)

From (2.1)-(2.10), we can easily obtain the following steady state equations:

( )

0,0 0,1 0= −βP +P 0 , (2.11)

( )

β

( )

₊

( ) ( )

− d _0, = − _0, + _{0, 1} 0 , 1≤ ≤ , duP n u P s un P n s u n F (2.12)

( )

+

( ) ( )

− d _0, = _{0, 1} 0 , + ≤ ≤1 −1, duP n u P n s u F n K (2.13)

( )

0, 1, 1 d _, duP K u λP K− u − = (2.14)

( )

1,0 0,0 1,1 0= −λP +βP +P 0 , (2.15)

( )

( ) ( )

1,1 1,1 0,1 1,0 1,2 d 0 duP u λP u βP s u λP s u P s u − = − + + + , (2.16)

( )

λ

( )

β

( )

λ ₋

( )

₊

( ) ( )

− = − + + + , ≤ ≤ 1, 1, 0, 1, 1 1, 1 d ₀ d 2 , n n n n n P u P u P s u P u P s u u n F (2.17)

( )

λ

( )

λ ₋

( )

₊

( ) ( )

− d _1, = − _1, + _{1, 1} + _{1, 1} 0 , + ≤ ≤1 −2, duP n u P n u P n u P n s u F n K (2.18)

( )

1, 1 1, 1 1, 2 d . duP K− u λP K− u λP K− u − = − + (2.19) Further define

( )

* 0 d 0 d , u u S θ =

∫

∞e−θ S u =

∫

∞e−θ s u u

( )

* 0,n ₀ u 0,n d , P θ =

∫

∞ −e θ P u u

( )

* 1,n ₀ u 1,n d , P θ =

∫

∞ −e θ P u u

( )

* 0,n 0,n 0 ₀ 0,n d , P =P =

∫

∞P u u

( )

* 1,n 1,n 0 ₀ 1,n d , P =P =

∫

∞P u u

( )

*

( )

0, 0, 0, 0 d 0 , u n n n e P u u P P u θ _θ _θ ∞ − ∂ ₌ ₋ ∂

∫

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and

( )

*

( )

1, 1, 1, 0 d 0 . u n n n e P u u P P u θ _θ _θ ∞ − ∂ ₌ ₋ ∂

∫

Therefore, we take the LST on both sides of (2.12)-(2.14) and (2.16)-(2.19). It yields

( )

( ) ( )

( )

θ θ β θ ₊ θ − * = − * + * − , ≤ ≤ 0,n 0,n 0, 1n 0 0,n 0 1 , P P S P S P n F (2.20)

( )

( ) ( )

( )

θ θ ₊ θ − * = * − , + ≤ ≤ − 0,n 0, 1n 0 0,n 0 1 1, P P S P F n K (2.21)

( )

* * 0,K 1,K 0,K 0 , P P P θ θ λ θ − = − (2.22)

(

) ( )

* *

( )

*

( )

( ) ( )

*

( )

1,1 0,1 1,0 1,2 0 1,1 0 P P S P S P S P λ θ− θ =β θ +λ θ + θ − , (2.23)

(

λ θ−

)

( )

θ =β

( )

θ +λ ₋

( )

θ + ₊

( ) ( )

θ −

( )

, ≤ ≤ * * * * 1, 0, 1, 1 1, 1 0 1, 0 2 , n n n n n P P S P P S P n F (2.24)

(

λ θ−

)

*

( )

θ =λ *₋

( )

θ + ₊

( ) ( )

* θ −

( )

, + ≤ ≤ − 1,n 1, 1n 1, 1n 0 1,n 0 1 2, P P P S P F n K (2.25)

(

)

*

( )

*

( )

1,K 1 1,K 2 1,K 1 0 . P P P λ θ− ₋ θ =λ ₋ θ − ₋ (2.26) 2.3.1 Recursive methods

The recursive method is developed to obtain P_0,*_n

( )

0 and P_1,*_n

( )

0 . Our solution algorithm will first obtain P_0,_n

( ) (

0 1≤ ≤n K

)

which are then used for finding P_0,*_n

( )

0 .

Using (2.11) and setting θ =0 in (2.20) and (2.21), we get

( )

βζ − = =

∑

1 ≤ ≤ 0, 0, 0 0 n , 1 , n i i P P n K where ζ = ⎨⎧ ≤ ≤ −_{≤ ≤} ⎩ , 0 1, , , n n n F F F n K (2.27) and

( )

βϕ

( )

+ = − + , ≤ ≤ − 0, 1n 0 n F, 0,n 0,n 0 1 1, P P P n K where ϕ = ⎨⎧ , ≤ ≤ ⎩ , 1 1 , 0, otherwise. n F n F (2.28)

Using (2.28) in (2.20) and (2.21), we get

( )

θ

( )

θ

( )

θ − = , ≤ ≤ − * * 0, 0, 1 0 1 1. n n S P P n K (2.29)

Taking lim_θ_→₀ in (2.29) and using L'Hospital'sˆ rule once gives

( )

=

( )

, ≤ ≤ −

*

0,n 0 1 0,n 0 1 1,

P s P n K (2.30)

where s₁ = −S* 1( )

( )

0 is the mean service time. Using (2.27) in (2.30), we have

( )

=φ , ≤ ≤ −

*

0,n 0 n 0,0 1 1

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where

(

)

ζ φ β β − = ⎧⎪ = ⎨ + ≤ ≤ ⎪⎩ 1 1 1 1, 0, 1 n , 1 . n n s s n K (2.32)

Thus, P_0,1*

( )

0 , 0 ,..., 0P_0,2*

( )

P_0,*_K₋₁

( )

can be obtained by using (2.31).

Next, we derive the expressions of P_1,_n

( ) (

0 1≤ ≤n K

)

in terms of P and _1,0

0,0

P . Using (2.31) in (2.23)-(2.24) and then setting θ λ= in (2.23)-(2.26), we finally obtain

( )

* * 1,1 1 0,0 1,0 1,2 * 0 0 P P S P S , P S βφ λ λ λ λ − − = (2.33)

( )

βϕ φ

_{( )}

( )

λ λ

( )

λ λ − + − − = 1, , 0,0 * 1, 1* ≤ ≤ − 1, 1 * 0 0 n n F n n , 2 2, n P P S P P n K S (2.34)

( )

*

( )

1,K 1 0 1,K 2 . P − =λP − λ (2.35) To obtain P_{1, 1}*_n−

( ) (

λ 1≤ ≤n K− in (2.34)-(2.35), using (2.31) in (2.23)-(2.24) 1

)

again, differentiating (2.23)-(2.26)

(

l −1

)

times with respect to θ and setting θ λ= , we finally get ( )−

_{( )}

_λ _{= −} ( )

( )

λ _⎡_λ ₊_βφ ₊_λ

_{( )}

_⎤ ₌ ₋ ⎣ ⎦ * * 1 1,0 1 0,0 1,2 1,1 0 , 1,..., 2 l l S P P P P l K l , (2.36) ( )−

_{( )}

_λ

_{( )}

( )

_{( )}

_λ _{βϕ φ} ( )

_{( )}

_λ _λ ( )

_{( )}

_λ + − ⎡ ⎤ = − _⎢ + + _⎥ ⎣ ⎦ ≤ ≤ − = − − * 1 * * * 1, 1 , 0,0 1, 1, 1 1 ₀ _, 2 2, 1,..., 1, l l l l n n F n n n P P S P S P l n K l K n (2.37)

( )

* 1( )

( )

* 1,K 1 1,K 2 P ₋ λ = −λP ₋ λ , (2.38)

where P_1,* 0_n( )

( )

λ =P_1,*_n

( )

λ and S*( )l

( )

θ _{= ⎣}⎡

(

d dl θl

)

S*

( )

θ ⎤_⎦ denotes the lth derivative of S*

( )

θ .

Solving (2.36)-(2.38) recursively, we obtain

( )

λ

( )

λ ζ β φ

( )

λ

( )

λ

( )

λ λ − + − + + = = = − − − ≤ ≤ −

∑

A

∑

A A * * 1 1 * * 1, 1,0 0,0 1, 1 1 1 0 , 1 1, n n n i i n i n n i i i S S P S P P P n K (2.39) where

( )

λ λ λ ⎧ − ⎪− ≤ ≤ − = ⎨ ⎪ ⎩ A * * , 1 1, ! 0, otherwise. n n n S n K n S (2.40)

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( )

_{( )}

( )

ζ λ β − φ ϕ φ λ − − − − + = − − − − − = = + + ⎡ ⎤ − + ≤ ≤ − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

∑

A A A 2 2 1, _* 1, 1 1 1, 1 1 1 1, 1 0,0 2 1,0 1 1 0 0 0 n , 3 1. n n n n i i i n i i n F n n i P P P S P P n K (2.41) We further define { } τ τ τ τ τ τ τ τ τ κ κ κ ≤ ≤ + + + = ∈ ⎧ ⎪ = ⎪ ⎪ Ψ =_⎨ = − ⎪ ⎪ ⎪⎩

∑

" " " " 1 2 1 2 1 2 1 , , , 1,2, , 1, 0, , 1, 2,..., 3, 0, otherwise, k k k n k n n n n n K (2.42) where

( )

λ κ ⎧ ₊ ₌ ⎪ ⎪⎪ =_⎨ = − ⎪ ⎪ ⎪⎩ A A " 1 * 1 , 1, , 2,3, , 3, 0, otherwise. n n n S n K (2.43)

Remark: The representative meaning of the above formulation (2.42) is to sum up all possible products of k κ in which the total of subscript values of s κ equals n. We give an easily understood example for n = 4:

4 4 3 1 2 2 1 3 1 1 2 1 2 1 2 1 1 1 1 1 1 2 2 4 4 3 1 2 1 2 1 2 3 . κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ κ Ψ = + + + + + + + = + + + +

Using (2.42) and (2.43) to solve (2.41) recursively, and including (2.15) and (2.33), we finally get

( )

1,1 0 1 1,0 1 0,0, P =A P +B P (2.44)

( )

−

( )

= ⎡ ⎤ =

∑

Ψ _⎣ + _⎦ ≤ ≤ − 1, 1,0 0,0 2 0 n , 2 1, n n i i P A i P B i P n K (2.45) where

( )

_{( )}

( )

λ λ λ λ λ ₋ = ⎧ ⎪ ⎡ ₋ ⎤ ⎪ =⎨ ⎢ ⎥ = ⎢ ⎥ ⎪ ⎣ ⎦ ⎪ _{≤ ≤} ₋ ⎩ A * * 2 , 1, 1 , 2, , 3 1, n n S A n n S n K (2.46)

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( )

_{( )}

( )

ζ β ϕ φ λ β λ β − _{− −}φ βϕ ₋ φ ₋ = ⎧ ⎪ − = ⎪ ⎪ _⎡ ₊ _⎤ ⎪ = −⎨ ⎢ ⎥ = ⎢ ⎥ ⎪ ⎣ ⎦ ⎪ ⎪ ₋ _{≤ ≤} ₋ ⎪⎩

∑

A 2 * 1, 1 * 1 1, 1 1 , 1, 1 , 2, , 3 1. n F n i i n F n i n S B n n S n K (2.47)

Substituting (2.45), (2.44), and (2.35) into (2.39) finally yields

(

) ( )

(

_{( )}

) ( )

(

) ( )

(

_{( )}

) ( )

ζ β φ λ λ λ − − + − − − − − = = = = − + − − − − = = = ⎡ _Ψ _{− −} ⎤ −⎢ Ψ − + + + ⎥ ⎢ ⎥ ⎣ ⎦ = ⎡ Ψ − − ⎤ Ψ − + + + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

∑

A A A A 2 2 1 1 1 * 1 1 2 2 1 1,0 ₂ ₁ ₁ 0,0 1 _* 2 1 2 2 1 1 1 1 K K i K K i K i i i j i i K i K K i K i j i K i B i i j B j S P P K i A i i j A j S . (2.48)

Finally, we develop the steady-state probabilities P_1,*_n

( )

0 in terms of P . Setting _0,0

0 θ = in (2.23)-(2.26) we have

( )

β φζ

( )

λ ₌ + ⎡ ⎤ = ⎢ + ⎥ ≤ ≤ − ⎢ ⎥ ⎣

∑

⎦ * 1, 0,0 1, 1 0 1 0 n 0 , 0 2, n i n i P P P n K (2.49)

( )

* 1, 1 0,0 0 0 F . K i i P β φP λ − = =

∑

(2.50)

As P_1,1

( )

0 , 0 ,..., 0P_1,2

( )

P_1,_K−₁

( )

and P1,0 are known, P1,1*

( )

0 , 0 ,P1,2*

( )

− * 1, 1

..., 0P _K can be determined recursively using (2.49) and (2.50) in terms of P . _0,0

Now the only unknown quantity is P_0,*_K

( )

0 which can be obtained from (2.22). To find it, differentiating (2.22) with respect to θ and setting θ =0, we have

( )

* 1( )

( )

*

0,K 0 1,K 1 0 .

P = −λP ₋ (2.51)

To find λP_1,* 1_K( )₋₁

( )

0 , differentiating (2.23)-(2.26) with respect to θ and setting

0 θ = , we finally obtain ( )

_{( )}

* 1( )

( )

* 1( )

( )

* 1( )

( )

* 1 1,1 1 0,0 1,0 1,2 1,1 0 0 0 0 0 P P S P S P S , P βφ λ λ + + + = (2.52) ( )

_{( )}

βϕ φ ( )

( )

λ ( )

( )

λ + − + + + = ≤ ≤ − * 1 * 1 * 1 * 1 1, , 0,0 1, 1 1, 1 1, 0 0 0 0 0 , 2 2, n n F n n n n P P S P P S P n K (2.53) ( )

_{( )}

* 1( )

( )

* 1 1, 1 1, 2 1, 1 0 0 K K . K P P P λ λ − − − + = (2.54)

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As P_1,1* 1( )

( )

0 is known completely from (2.52), the values P_1,* 1_n( )

( )

0 for 2,3,..., 1

n= K− can be found recursively from (2.53) and (2.54). Therefore we obtain ( )

_{( )}

_β ( )

_{( )}

_φ ( )

_{( )}

_λ ( )

_{( )}

λ − − − = = = ⎡ ⎤ = _⎢ + + + _⎥ ⎣

∑

⎦ 1 1 * 1 * 1 * 1 * 1 1, 0,0 1, 1,0 1, 1 1 1 2 1 0 K _i 0 F _n 0 K _i 0 0 . K i i i P P S P S P P S (2.55)

Substituting (2.55) into (2.51), we have

( )

1 * 1( )

( )

* 1( )

( )

1

( )

* 1( )

( )

* 0, 1, 0,0 1, 1,0 1 1 2 0 K 0 F 0 K 0 0 . K i n i i i i P − P βS φ P S − P λP S = = = ⎡ ⎤ = −_⎢ + + + _⎥ ⎣

∑

⎦ (2.56)

So P0,1*

( )

0 , 0 ,..., 0P0,2*

( )

P0,*K

( )

is known in terms of P0,0 , which can be

determined using the normalizing condition

1 0, 1, 0 0 1. K K i i i i P − P = = + =

∑

(2.57)

To demonstrate the working of the recursive method, we first describe the solution algorithm for calculating the steady state probabilities, P_0,*_n

( ) (

0 0≤ ≤n K

)

and

( ) (

)

*

1,n 0 0 1

P ≤ ≤n K− . Next, to illustrate the solution algorithm, we provide three simple examples where the service time distributions are exponential, k-stage Erlang, and deterministic, respectively.

2.3.2 The solution algorithm

Let F be the threshold, K be the maximum capacity of the system, and let

( )

_{( )}

* l

S θ be the l-th derivative of S*

( )

θ , where l =1, 2,...,K. We set the values of

F, K, and the LST expression of the service time distribution, namely S*

( )

θ . The steps of the solution algorithm are stated as follows:

Step 1. For each n=0, 1,...,K , compute φ_n using (2.32).

Step 2. For each n=1, 2,...,K− , compute 1 *

( )

0,n 0

P using (2.31) in terms of

0,0

P .

Step 3. Compute A_n

(

1≤ ≤n K−2

)

and κ_n

(

1≤ ≤n K−3

)

using (2.40) and (2.43), respectively.

Step 4. For each n=0, 1,...,K− , compute 3 Ψ using (2.42). _n

Step 5. For each n=1, 2,...,K − , compute 1 A n

( )

and B n

( )

using (2.46) and (2.47).

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Step 6. For each n=1, 2,...,K− , compute 1 P_1,_n

( )

0 using (2.44) and (2.45) in terms of P and 1,0 P . 0,0

Step 7. Compute P using (2.48) in terms of _1,0 P . Thus _0,0 P1,n

( )

0

(

1≤ ≤n K−1

)

are achieved from Step 6.

Step 8. For each n=1, 2,...,K− , compute 1 *

( )

1,n 0

P using (2.49) and (2.50) in terms of P . _0,0

Step 9. For n K= , compute P_0,*_n

( )

0 using (2.56) in terms of P . _0,0

Step 10. Determine P using (2.57). Thus 0,0 P0,*n

( ) (

0 1, n= 2,...,K

)

are achieved from Steps 2 and 9, and *

( ) (

)

1,n 0 0, 1,..., 1

P n= K− are achieved from

Steps 7 to 8.

2.4 Simple Examples

We use the solution algorithm to illustrate a recursive method. We provide three simple examples for three different service time distributions such as exponential, 3-stage Erlang, and deterministic, respectively.

Example 1 (For M/M/1 queue). We set the mean service time s₁=1 μ , where μ

is the service rate. Assume that F = and 1 K = . In this case, we have 4

( )

* S θ μ μ θ = + .

Step 1. For each n =0, 1,...,4, compute φ_n. Using (2.32), we obtain

0 1

φ = , φ₁= −

(

1 α α

)

, and φ₂ =φ₃ =φ₄ = −

(

1 α α

)

2, where α μ μ β=

(

+

)

.

Step 2. For each n =1, 2, 3, compute P_0,*_n

( )

From (2.31), we finally get

( )

* 0,1 0 1 0,0 1 0,0 P φ P α P α − = = ,

( )

* * 0,2 0,3 2 0,0 2 0,0 1 0 0 P P φ P α P α − = = = .

Step 3. For each n =1, 2 , compute A_n and κ_n using (2.40) and (2.43), respectively.

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σ μ λ= .

For each n =1, we find from (2.43) that κ1 = + +

(

1 σ σ2

)

σ

(

1+σ

)

.

Step 4. For each n =0, 1, compute Ψ . _n

It implies from (2.42) that Ψ = and ₀ 1

(

2

)

(

)

1 1 σ σ σ 1 σ

Ψ = + + + .

Step 5. For each n =1, 2, 3, compute A n

( )

and B n

( )

. Using (2.46) and (2.47), it follows that

( )

1 A =μ σ , A

( )

2 =μ σ2 , and A

( )

3 = −μ σ

(

1+σ

)

.

( )

1

(

1

)

B α μ α − = − , B

( )

2

(

α σ

)(

1₂ α μ

)

σα + − = − , and

( )

(

)

(

)

2 2 1 3 1 B α μ σ α − = − + .

Step 6. For each n =1, 2, 3, compute P_1,_n

( )

0 using (2.44) and (2.45) in terms of

1,0

P and P . _0,0

It yields from (2.44) and (2.45) that

( )

1,1 0 1 1,0 1 0,0 P =A P +B P ,

( )

1,2 0 0 2 1,0 2 0,0 P = Ψ ⎡_⎣A P +B P ⎤_{⎦ ,}

( )

1,3 0 1 2 1,0 2 0,0 0 3 1,0 3 0,0 P = Ψ _⎣⎡A P +B P _⎦⎤+ Ψ ⎡_⎣A P +B P ⎤_{⎦ .}

Step 7. Compute P using (2.48) in terms of _1,0 P . Thus _0,0 P_1,_n

( ) (

0 1≤ ≤n 3

)

are achieved from Step 6.

From (2.48), we finally have

(

)

(

2 3

)

1,0 ₂ 0,0 1 P σ α α σ σ σ P α − + + + = ,

(

P_1,0*

( )

0 =P_1,0

)

,

( )

(

)

(

)

2 1,1 ₂ 0,0 1 1 0 P σμ α σ σ P α − + + = ,

( )

(

)(

)

1,2 ₂ 0,0 1 1 0 P σμ α σ P α − + = ,

( )

(

)

1,3 ₂ 0,0 1 0 P σμ α P α − = .

Step 8. For each n =1, 2, 3, compute P_1,*_n

( )

0 using (2.49) and (2.50) in terms of P . _0,0

Using (2.49) and (2.50) yields

( )

(

)

(

2

)

* 1,1 2 0,0 1 1 0 P σ α σ σ P α − + + = , P_1,2*

( )

0 σ

(

1 α

)(

₂ 1 σ

)

P_0,0 α − + = ,

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and P_1,3*

( )

0 σ

(

1₂α

)

P_0,0

α

−

= .

Step 9. For n =4, compute *

( )

0,n 0

P using (2.56) in terms of P . _0,0

Using (2.56), it follows that

( ) (

)

* 0,4 2 0,0 1 0 P α P α − = .

Step 10. Determine P using (2.57). Thus 0,0 P0,*n

( ) (

0 0, n = 1,...,4

)

are achieved from Steps 2 and 9, and P1,*n

( ) (

0 0, n = 1, 2, 3

)

are achieved from Steps 7 to 8.

(

) (

)

(

)

(

)

2 0,0 ₂ ₂ ₃ 1 3 1 1 3 3 2 P α α α α α σ α α σ σ σ = + − + − + − + + + + .

It is to be noted that these results are the same as those given in Gupta [12, p1006].

Example 2 (For M/E3/1 queue). The 3-stage Erlang distribution is made up of three independent and identical exponential stages, each with mean 1 3μ. We set the mean service time s₁ =1 μ , 1F = , and K =3. In this case, we have

( )

3 * 3 3 S θ μ μ θ ⎛ ⎞ = ⎜ ₊ ⎟ ⎝ ⎠ .

Step 1. For each n =0, 1,...,3, compute φ_n. From (2.32), we finally obtain

0 1

φ = , φ₁=3 1

(

−γ γ

)

, and φ₂ =φ₃ =3 1

(

−γ

)(

3 2− γ γ

)

2, where γ =3μ

(

3μ β+

)

.

Step 2. For each n =1, 2, compute P_0,*_n

( )

From (2.31), it follows that

( )

* 0,1 0 1 0,0 31 0,0 P φ P γ P γ − = = ,

( )

(

)(

)

* 0,2 2 0,0 2 0,0 1 3 2 0 3 P φ P γ γ P γ − − = = .

Step 3. For each n =1, compute A_n.

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Step 4. For each n =0, compute Ψ . _n It implies from (2.42) that Ψ = . 0 1

Step 5. For each n =1, 2, compute A n

( )

and B n

( )

. It yields from (2.46) and (2.47) that

( )

1 3 A = μ τ and A

( )

2 =3 1 3μ

(

+ τ +3τ2

)

τ4.

( )

1 3

(

1

)

B γ μ γ − = − and

( )

(

)

(

) (

)

2 3 3 2 1 3 3 3 2 1 2 3 B α τ τ τ γ γ μ τ γ ⎡ ₊ ₊ ₊ ₋ ⎤ ₋ ⎣ ⎦ = − .

Step 6. For each n =1, 2, compute P_1,_n

( )

0 using (2.44) and (2.45) in terms of P _1,0

and P . _0,0

From (2.44) and (2.45), we find that

( )

1,1 0 1 1,0 1 0,0 P =A P +B P ,

( )

1,2 0 0 2 1,0 2 0,0 P = Ψ ⎡_⎣A P +B P ⎤_{⎦ ,}

Step 7. Compute P using (2.48) in terms of _1,0 P . Thus _0,0 P1,n

( ) (

0 1≤ ≤n 2

)

are achieved from Step 6.

It implies from (2.48) that

(

) (

)(

)

(

)

(

)

3 2 1,0 ₂ ₂ 0,0 1 1 3 2 1 4 6 1 4 6 P τ γ τ τ γ γ τ τ P τ τ γ ⎡ ⎤ − _⎣ + − + + + _⎦ = + + ,

(

( )

)

* 1,0 0 1,0 P =P ,

( )

(

)(

)

(

)

3 1,1 ₂ ₂ 0,0 1 1 3 2 0 3 1 4 6 P τ μ τ γ γ P τ τ γ + − − = + + ,

( )

(

)(

)

(

)

3 1,2 ₂ ₂ 0,0 1 3 2 0 9 1 4 6 P τ μ γ γ P τ τ γ − − = + + .

( )

0 using (2.49) and (2.50) in terms of P . _0,0

Using (2.49) and (2.50) yields

( )

(

)(

)

(

)

(

)

2 * 1,1 ₂ ₂ 0,0 1 1 3 2 1 3 3 0 1 4 6 P τ τ γ γ τ τ P τ τ γ + − − + + = + + , and

( )

(

)(

)

(

)

2 * 1,2 ₂ 0,0 1 3 2 1 3 3 0 P τ γ γ τ τ P γ − − + + = .

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Step 9. For n =3, compute *

( )

0,n 0

P using (2.56) in terms of P . _0,0

It follows from (2.56) that

( )

(

)(

)

(

)

(

)

2 * 0,3 ₂ ₂ 0,0 1 3 2 3 10 10 0 1 4 6 P γ γ τ τ P τ τ γ − − + + = + + .

Step 10. Determine P using (2.57). Thus _0,0 P_0,*_n

( ) (

0 0, n = 1,...,3

)

are achieved from Steps 2 and 9, and P_1,*_n

( ) (

0 0, n = 1, 2

)

are achieved from Steps 7 to 8.

(

)

(

)

(

)

(

)

{

(

)(

)

(

)

}

2 2 0,0 2 2 2 1 2 3 4 5 1 4 6 1 4 6 1 9 1 3 2 3 11 14 6 4 . P τ τ γ τ τ γ γ γ γ γ τ τ τ τ τ − = + + × ⎡ ⎤ + + _⎣ + − − _⎦ + − − + + + + +

Example 3 (For M/D/1 queue). We set the mean service time s₁=1 μ , F = , 1 and K =3. In this case,

( )

*

S θ =e−θ μ.

Step 1. For each n =0, 1,...,3, compute φ_n. Using (2.32) yields

0 1

φ = , φ₁= −

(

1 α α

)

, and φ₂ =φ₃ = −

(

1 α α

)

2, where α μ μ β=

(

+

)

.

( )

Using (2.31), we finally get

( )

* 0,1 0 1 0,0 1 0,0 P φ P α P α − = = ,

( )

* 0,2 2 0,0 2 0,0 1 0 P φ P α P α − = = .

Step 3. For each n =1, compute A_n.

From (2.40), we find that A₁ = −ρ, where ρ λ μ= .

Step 4. For each n =0, compute Ψ . _n It implies from (2.42) that Ψ = . ₀ 1

Step 5. For each n =1, 2, compute A n

( )

and B n

( )

. From (2.46) and (2.47), it follows that

M/G/1/K與G/M/1/K排隊含啟動時間的F方策與N方策之相互關係

國

立

交

通

大

學

工業工程與管理學系

博

士

論

文