A recursive method for the F-policy G/M/1/K queueing system with an exponential startup time

A recursive method for the F-policy G/M/1/K queueing

system with an exponential startup time

Kuo-Hsiung Wang

, Ching-Chang Kuo

, W.L. Pearn

a

Department of Applied Mathematics, National Chung-Hsing University, Taichung 402, Taiwan b

Department of Industrial Engineering and Management, National Chiao Tung University, Hsin Chu 30050, Taiwan Received 1 May 2006; received in revised form 1 February 2007; accepted 28 February 2007

Available online 13 March 2007

Abstract

This paper deals with the optimal control of a finite capacity G/M/1 queueing system combined the F-policy and an exponential startup time before start allowing customers in the system. The F-policy queueing problem investigates the most common issue of controlling arrival to a queueing system. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining interarrival time, to develop the steady-state probability distribution of the number of customers in the system. We illustrate a recursive method by presenting three simple examples for exponential, 3-stage Erlang, and deterministic interarrival time distributions, respectively. A cost model is developed to determine the optimal management F-policy at minimum cost. We use an efficient Maple computer program to determine the optimal operating F-policy and some system performance measures. Sensitivity analysis is also studied.

Ó 2007 Elsevier Inc. All rights reserved.

Keywords: F-Policy; G/M/1/K queue; Recursive method; Server startup; Supplementary variable

1. Introduction

We use a supplementary variable technique to analyze the optimal control of the F-policy G/M/1/K

queue-ing system where the server needs a startup time before start allowqueue-ing customers in the system and K <1

denotes the maximum capacity of the system. The method of controlling arrivals focuses on reducing the num-ber of customers in the system. The model proposed in this paper 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[1]. However, steady-state analytical solutions of the F-policy queueing

systems with interarrival times or service times distribution of the general type have not been found. It is extre-mely difficult, if not possible, to obtain the explicit expressions for the steady-state probability distribution of

0307-904X/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2007.02.023

* _{Corresponding author.}

E-mail address:[email protected](K.-H. Wang).

Available online at www.sciencedirect.com

Applied Mathematical Modelling 32 (2008) 958–970

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 [2]first introduced the supplementary variable technique. Basis on this technique, Gupta and

Rao [3,4] 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 [5]. The extension of this model can be

referred to Bell [6,7], Heyman [8], Kimura[9], Teghem[10], Wang and Ke [11], and others. Wang and Ke

[11]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 [12] 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. 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[13]first

studied the N-policy M/M/1 queueing system with an exponential startup time. Borthakur et al.[14]extended

Baker’s model to the general startup time. The N-policy M/G/1 queueing system with startup time was

inves-tigated by several authors such as Medhi and Templeton[15], Takagi[16], Lee and Park[17], Hur and Paik

[18], Krishna et al.[19], and so on. Ke[20]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 expo-nential startup time. In the second category of controlling the arrivals, the analytical developments for con-trolling 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 con-centrates 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[1]. Through a series

of propositions, the relationship between the operating N-policy and the operating F-policy are established by Gupta [1].

Practically, the memoryless property of the arrival (input) process does not always meets the needs of appli-cations 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 situa-tions satisfy the assumpsitua-tions of Markovian condisitua-tions for service time. Hence, we may consider inevitably to analyze the F-policy G/M/1/K queueing system.

In Section2, the queueing model is briefly described. Practical justification of the model is also included.

Section3provides a recursive method using the supplementary variable technique and treating the

supplemen-tary variable as the remaining interarrival time, to obtain the steady-state probability distributions of the num-ber of customers in the F-policy G/M/1/K queueing system. We illustrate the solution algorithm by presenting three simple examples for three different interarrival time distributions: exponential (denoted M), 3-stage

Erlang (denoted E3), and deterministic (denoted D). In Section4, various system performance measures are

presented. The total expected cost function per unit time for the F-policy G/M/1/K queueing system with

startup times is developed in Section 4. Numerical and comparative results are shown in Section5. Finally,

Section6 consists of some concluding remarks.

2. Description of the system

We consider the category of controlling the arrivals to the F-policy G/M/1/K queueing system with expo-nential startup time. It is assumed that the times elapsing between successive arrivals are independent and identically distributed (i.i.d) random variables having general distribution A(v) (v P 0), a probability density

function a(v) (v P 0) and mean interarrival time b1. The service times of the customers are independent

arriving at the server form a single waiting line and are served in the order of their arrivals; that is, according to the first-come, first-served (FCFS) discipline. Suppose that the server can serve only one customer at a time, and that the service is independent of the arrival of the customers. 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[1]first intro-duced the concept of a F-policy. The definition of a F-policy is described as follows: When the number of cus-tomers in the system reaches its capacity K (i.e. the system becomes full), no further arriving cuscus-tomers are allowed to enter the system until there are enough customers in the system have been served so that the num-ber of customers in the system decreases to a threshold value Fð0 6 F < K  1Þ. At that time, the server needs to take an exponential startup time with parameter b to start allowing customers in the system. Thus, the sys-tem operates normally until the number of customers in the syssys-tem reaches its capacity at which time the above process is repeated all over again.

2.1. Practical justification of the model

A number of practical problems arise which may be formulated as one in which the server requires take 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, messages 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 a specified 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 Thanksgiving day, the entrance ramps along the highway will be con-trolled 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.

2.2. Notation

The following notation and definitions are used throughout the paper:

F threshold level

K system capacityðK > F þ 1Þ

A interarrival time random variable

V remaining interarrival time random variable

A(v) distribution function (d.f.) of A

a(v) probability density function (p.d.f.) of A

aðhÞ Laplace–Stieltjes transform (LST) of A

aðlÞ_{ðhÞ lth order derivative of a}_{ðhÞ with respect to h}

P0;0ðtÞ probability of no customers in the system at time t when the arrivals are not allowed to enter the

sys-tem

P_0;nðtÞ probability of n customers in the system at time t when the arrivals are not allowed to enter the sys-tem, where n¼ 1; 2; . . . ; K

P1;0ðtÞ probability of no customers in the system at time t when the arrivals are allowed to enter the system

P1;nðtÞ probability of n customers in the system at time t when the arrivals are allowed to enter the system,

where n¼ 1; 2; . . . ; K  1 3. Steady-state results

The state of the system at time t is given by

N(t) number of customers in the system, and

Let us define P0;nðv; tÞdv ¼ PrfN ðtÞ ¼ n; v < V ðtÞ 6 v þ dvg; vP0; n¼ 0; 1; . . . ; F ; P1;nðv; tÞdv ¼ PrfN ðtÞ ¼ n; v < V ðtÞ 6 v þ dvg; vP0; n¼ 0; 1; . . . ; K  1; P0;nðtÞ ¼ Z 1 0 P0;nðv; tÞdv; n¼ 0; 1; . . . ; F ; P1;nðtÞ ¼ Z 1 0 P1;nðv; tÞdv; n¼ 0; 1; . . . ; K  1:

In steady state, we define P0;n¼ lim t!1P0;nðtÞ; n¼ 0; 1; . . . ; K; P1;n¼ lim t!1P0;nðtÞ; n¼ 0; 1; . . . ; K  1; P0;nðvÞ ¼ lim t!1P0;nðv; tÞ; n¼ 0; 1; . . . ; F ; P1;nðvÞ ¼ lim t!1P1;nðv; tÞ; n¼ 0; 1; . . . ; K  1

and further define

P0;nðvÞ ¼ P0;naðvÞ; n¼ 0; 1; . . . ; F : ð1Þ

For the F-policy G/M/1/K queueing system with server startup, we can easily obtain the steady-state equa-tions as follows: 0¼ bP0;0þ lP0;1; ð2Þ 0¼ ðb þ lÞP0;nþ lP0;nþ1; 1 6 n 6 F ; ð3Þ 0¼ lP0;nþ lP0;nþ1; F þ 1 6 n 6 K  1; ð4Þ 0¼ lP0;Kþ P1;K1ð0Þ; ð5Þ  d dvP1;0ðvÞ ¼ bP0;0aðvÞ þ lP1;1ðvÞ; ð6Þ  d dvP1;nðvÞ ¼ lP1;nðvÞ þ bP0;naðvÞ þ P1;n1ð0ÞaðvÞ þ lP1;nþ1ðvÞ; 1 6 n 6 F ; ð7Þ  d dvP1;nðvÞ ¼ lP1;nðvÞ þ P1;n1ð0ÞaðvÞ þ lP1;nþ1ðvÞ; Fþ 1 6 n 6 K  2; ð8Þ  d dvP1;K1ðvÞ ¼ lP1;K1ðvÞ þ P1;K2ð0ÞaðvÞ: ð9Þ

We introduce the following Laplace–Stieltjes transforms (LST): aðhÞ ¼ Z 1 0 ehvdAðvÞ ¼ Z 1 0 ehvaðvÞdv; P_i;nðhÞ ¼ Z 1 0 ehv_P i;nðvÞdv; i¼ 0; 1; Pi;n¼ Pi;nð0Þ ¼ Z 1 0 Pi;nðvÞdv; i¼ 0; 1; Z 1 0 ehv o ouPi;nðvÞdv ¼ hP  i;nðhÞ  Pi;nð0Þ; i¼ 0; 1:

Taking the LST on both sides of (6)–(9), it implies that  hP

1;0ðhÞ ¼ bP0;0aðhÞ þ lP1;1ðhÞ  P1;0ð0Þ; ð10Þ

ðl  hÞP

ðl  hÞP 1;nðhÞ ¼ lP  1;nþ1ðhÞ þ P1;n1ð0ÞaðhÞ  P1;nð0Þ; Fþ 1 6 n 6 K  2; ð12Þ ðl  hÞP 1;K1ðhÞ ¼ P1;K2ð0ÞaðhÞ  P1;K1ð0Þ: ð13Þ 3.1. Recursive method

Our main work is to develop the steady-state probabilities P_0;nð0Þ and P1;nð0Þ, where 1 6 n 6 K. Our

solu-tion algorithm will first obtain P_0;nð0Þ ð1 6 n 6 KÞ. Using(2)–(5)yields P_0;nð0Þ ¼ /nP0;0; 1 6 n 6 K; ð14Þ P_1;K1ð0Þ ¼ l/_Fþ1P0;0; ð15Þ where /n¼ 1; n¼ 0; b l 1þ b l  fn1 ; 1 6 n 6 K; 8 > < > : ð16Þ and fn¼ n; 0 6 n 6 F 1; F ; F 6 n 6 K:  Thus, P

0;1ð0Þ; P0;2ð0Þ; . . . ; P0;Kð0Þ can be obtained by using(14).

Next, we derive the expressions of P1;nð0Þ ð0 6 n 6 K  2Þ in terms of P0;0. Substituting (14), (15) into

(11)–(13)and then setting h¼ l in(11)–(13) we finally have P1;nð0Þ ¼ P_1;nþ1ð0Þ  lP_1;nþ2ðlÞ  bunþ1;F/nþ1P0;0aðlÞ a_ðlÞ ; 0 6 n 6 K 3; ð17Þ where un;F ¼ 1; 1 6 n 6 F ; 0; otherwise;  P1;K2ð0Þ ¼ l/_Fþ1 a_ðlÞP0;0: ð18Þ

To obtain P_1;nþ2ðlÞ ð0 6 n 6 K  3Þ in (17), substituting (14) and (18) into(11)–(13) then differentiating

(11)–(13)ðl  1Þ times with respect to h, and finally setting h ¼ l, we get Pðl1Þ_1;n ðlÞ ¼ 1 l½P1;n1ð0Þa ðlÞ_{ðlÞ þ bu} n;F/nP0;0aðlÞðlÞ þ lPðlÞ_1;nþ1ðlÞ; ð19Þ where 2 6 n 6 K 2, l ¼ 1; . . . ; K  n  1, P_1;K1ðlÞ ¼ l/Fþ1a ð1Þ_ðlÞ a_ðlÞ P0;0; ð20Þ where Pð0Þ_1;n ðlÞ ¼ P 1;nðlÞ.

Solving(19) and (20)recursively, we finally obtain

P_1;nðlÞ ¼ bunþ2;Fa _ðlÞ l XF i¼fnþ2 ‘ni1/iP0;0 aðlÞ l XK1 i¼nþ2 ‘ni1P1;i1ð0Þ; 2 6 n 6 K 2; ð21Þ

where ‘n¼ ðlÞ n aðnÞ_ðlÞ n!a_ðlÞ ; 1 6 n 6 K 1; 0; otherwise: 8 > < > : ð22Þ

Using(20) and (21)in(17), we obtain P1;nð0Þ ¼ P1;nþ1ð0Þ a_ðlÞ þ XK1 i¼nþ2 ‘_in1P_1;i1ð0Þ þ b u_nþ2;F X F i¼fnþ2 ‘_in1/i unþ1;F/nþ1 " # P0;0; 0 6 n 6 K 3: ð23Þ Further, let us define

Wn¼ 1; n¼ 0; P 16k6n P s1þs2þþsk¼n s1;s2;;sk2f1;2;;ng js1js2   jsk; n¼ 1; 2; . . . ; K  2; 0; otherwise; 8 > > > > < > > > > : ð24Þ where jn¼ 1 a_ðlÞþ ‘1; n¼ 1; ‘n; n¼ 2; 3; . . . ; K  2; 0; otherwise: 8 > > > < > > > : ð25Þ

Remark. The representative meaning of the above formulation(24)is to sum up all possible products of k js

in which the total of subscript values of j equals n. We may present an easily understood example for n¼ 4:

W4¼ j4þ j3j1þ j2j2þ j1j3þ j1j1j2þ j1j2j1þ j2j1j1þ j1j1j1j1¼ j4þ 2j3j1þ j22þ 3j 2

1j2þ j41: ð26Þ

Using(24) and (25)to solve(23)recursively, and including(18), we finally have P1;nð0Þ ¼ X Kn1 i¼1 WKni1KðK  i  1ÞP0;0; 0 6 n 6 K 2; ð27Þ where KðnÞ ¼ bu_nþ2;F P F i¼fnþ2 ‘_in1/i bunþ1;F/nþ1; 0 6 n 6 K 3; l/Fþ1 a_ðlÞ; n¼ K  2: 8 > > > < > > > : ð28Þ

Finally, we develop the steady-state probabilities P

1;nð0Þ in terms of P0;0. Setting h¼ 0 in (10)–(13)yields

P_1;nð0Þ ¼1 l P1;n1ð0Þ  b Xfn1 i¼0 /_iP0;0 " # ; 1 6 n 6 K 1: ð29Þ As P1;1ð0Þ; P1;2ð0Þ; . . . ; P1;K1ð0Þ are known, P1;1ð0Þ; P  1;2ð0Þ; . . . ; P 

1;K1ð0Þ can be solved recursively using(29)in

Now the only unknown quantity is P

1;0ð0Þ which can be determined from(10)–(13). To find it,

differenti-ating(10)–(13)with respect to h and then setting h¼ 0, we get

P_1;0ð0Þ ¼ bP0;0að1Þð0Þ  lPð1Þ1;1 ð0Þ; ð30Þ

Pð1Þ_1;n ð0Þ ¼P1;nþ bun;F/nP0;0a

ð1Þ_{ð0Þ þ lP}ð1Þ

1;nþ1ð0Þ þ P1;n1ð0Það1Þð0Þ

l ; 1 6 n 6 K 1: ð31Þ

The values Pð1Þ_1;nð0Þ for n ¼ 1; 2; . . . ; K  1 can be found recursively from(31). Therefore, we obtain P_1;0ð0Þ ¼  bað1Þ_ð0ÞX F i¼0 /_nP0;0þ X K1 i¼1 P1;iþ að1Þð0Þ XK2 i¼0 P1;ið0Þ " # : ð32Þ So P

1;0ð0Þ; P1;1ð0Þ; . . . ; P1;K1ð0Þ are known in terms of P0;0, which can be determined using the normalizing

condition XK i¼0 P0;iþ XK1 i¼0 P1;i ¼ 1: ð33Þ

3.2. The solution algorithm

The steps of the solution algorithm are stated as follows: Step 1. For n¼ 0; 1; . . . ; K, compute /n using(16).

Step 2. For n¼ 1; 2; . . . ; K, compute P

0;nð0Þ using(14)in terms of P0;0.

Step 3. Compute ‘n ð1 6 n 6 K  2Þ and jnð1 6 n 6 K  2Þ using(22) and (25), respectively.

Step 4. For n¼ 0; 1; . . . ; K  2, compute Wnusing(24).

Step 5. For n¼ 0; 1; . . . ; K  2, compute KðnÞ using (28).

Step 6. For n¼ 0; 1; . . . ; K  2, compute P1;nð0Þ using(27)in terms of P0;0.

Step 7. For n¼ 1; 2; . . . ; K  1, compute P

1;nð0Þ using(29)in terms of P0,0.

Step 8. Compute P1;0ð0Þ using(32)in terms of P0,0.

Step 9. Determine P0;0 using (33). Thus P0;nð0Þ ðn ¼ 1; 2; . . . ; KÞ are achieved from Step 2, and

P_1;nð0Þ ðn ¼ 0; 1; . . . ; K  1Þ are achieved from Steps 7 and 8. 3.3. Simple examples

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

Example 1 (For M/M/1/K queueing system). We set the mean interarrival time b1¼ 1=k, where k is the

interarrival rate. Assume that F ¼ 2 and K ¼ 5. In this case, we have

aðhÞ ¼ k

kþ h:

Step 1. For n¼ 0; 1; . . . ; 5, compute /n using(16).

Using(16), we have /0¼ 1; /1¼ 1 a a ; /2¼ 1 a a2 ; and /3¼ /4¼ /5¼ 1 a a3 ; where a¼ l=ðl þ bÞ:

Step 2. For n¼ 1; 2; . . . ; 5, compute P

0;nð0Þ using(14)in terms of P0,0.

Using(14), we finally obtain P_0;1ð0Þ ¼ /1P0;0¼ 1 a a P0;0; P  0;2ð0Þ ¼ /2P0;0¼ 1 a a2 P0;0;

P_0;3ð0Þ ¼ P 0;4ð0Þ ¼ P  0;5ð0Þ ¼ /3P0;0 ¼ 1 a a3 P0;0:

Step 3. For n¼ 1; 2; 3, compute ‘n and jnusing(22) and (24), respectively.

We first compute ‘n. For n¼ 1; 2; 3, using (22) yields ‘1¼ r=ð1 þ rÞ, ‘2¼ r2=ð1 þ rÞ2 and

‘3¼ r3=ð1 þ rÞ 3

, where r¼ l=k.

Next, we compute jn. For n¼ 1; 2; 3, using(24)yields

j1¼ ð1 þ r þ r2Þ=ð1 þ rÞ; j2¼ r2=ð1 þ rÞ 2

and j3¼ r3=ð1 þ rÞ

3

: Step 4. For n¼ 0; 1; 2; 3, compute Wnusing(23).

It follows from(23)that

W0¼ 1; W1¼ ð1 þ r þ r2Þ=ð1 þ rÞ; W2¼ 1 þ r2; and W3¼ ð1 þ r þ r2þ r3þ r4Þ=ð1 þ rÞ:

Step 5. For n¼ 0; 1; 2; 3, compute KðnÞ using(28).

From (28)we have Kð0Þ ¼ lð1  aÞ 2 ða þ ar þ r2_Þ a3_{ð1 þ rÞ} ; Kð1Þ ¼  lð1  aÞ2 a3 ; Kð2Þ ¼ 0; and Kð3Þ ¼ lð1  aÞð1 þ rÞ a3 :

Step 6. For n¼ 0; 1; 2; 3, compute P1;nð0Þ using(27)in terms of P0,0.

Using(27)yields P1;0ð0Þ ¼ ½W3Kð3Þ þ W2Kð2Þ þ W1Kð1Þ þ W0Kð0ÞP0;0¼ lð1  aÞðr2_{þ r}3_{þ r}4_{þ ar þ a}2_Þ a3 P0;0; P1;1ð0Þ ¼ ½W2Kð3Þ þ W1Kð2Þ þ W0Kð1ÞP0;0¼ lð1  aÞðr þ r2_{þ r}3_{þ aÞ} a3 P0;0; P1;2ð0Þ ¼ ½W1Kð3Þ þ W0Kð2ÞP0;0¼ lð1  aÞð1 þ r þ r2_Þ a3 P0;0; P1;3ð0Þ ¼ W3Kð3ÞP0;0¼ lð1  aÞð1 þ rÞ a3 P0;0:

Step 7. For n¼ 1; 2; 3; 4, compute P

1;nð0Þ using(29)in terms of P0,0.

It implies from(29)that P_1;1ð0Þ ¼rð1  aÞðr þ r 2_{þ r}3_{þ aÞ} a3 P0;0; P  1;2ð0Þ ¼ rð1  aÞð1 þ r þ r2_Þ a3 P0;0; P_1;3ð0Þ ¼rð1  aÞð1 þ rÞ a3 P0;0; and P  1;4ð0Þ ¼ rð1  aÞ a3 P0;0: Step 8. Compute P 1;0ð0Þ using(32)in terms of P0,0. Using(32)yields P 1;0ð0Þ ¼ rð1aÞðr2_þr3_þr4_þarþa2_Þ a3 P0;0.

Step 9. Determine P0,0 using (33). Thus P0;nð0Þ ðn ¼ 1; 2; . . . ; 5Þ are achieved from Step 2, and

P_1;nð0Þ ðn ¼ 0; 1; . . . ; 4Þ are achieved from Step 7 and Step 8. P0;0¼

a3

a3_{þ ð1  aÞð3 þ a þ a}2_{Þ þ rð1  aÞð3 þ a þ a}2_{þ ar þ 3 þ 3r þ 3r}2_{þ 2r}3_{þ r}4_Þ:

It is to be noted that these results are in accordance with the expressions given by Gupta[1, p. 1006].

Example 2. (For E3/M/1/K queueing system). Let k denote the interarrival rate. The 3-stage Erlang

distribution is made up of three independent and identical exponential stages, each with mean 1=3k. We set the

mean interarrival time b1¼ 1=k, F ¼ 1, and K ¼ 3. In this case, we have

aðhÞ ¼ 3k

3kþ h

 3

Step 1. For n¼ 0; 1; . . . ; 3, compute /nusing(16).

Using(16), we obtain

/0¼ 1; /1¼ 3ð1  cÞ=c; and /2¼ /3¼ 3ð1  cÞð3  2cÞ=c 2_;

where c¼ 3l=ð3l þ bÞ:

Step 2. For n¼ 1; 2; 3, compute P

0;nð0Þ using(14)in terms of P0,0. Using(14)yields P_0;1ð0Þ ¼ /1P0;0 ¼ 3 1 c c P0;0; P  0;2ð0Þ ¼ P  0;3ð0Þ ¼ /2P0;0¼ 3 ð1  cÞð3  2cÞ c2 P0;0:

Step 3. Compute ‘1and j1using(22) and (25), respectively.

Using(22)yields ‘1¼ 3s=ð1 þ sÞ, where s ¼ l=3k.

Form(25), we obtain j1¼ ð1 þ s þ 6s2þ 4s3þ s4Þ=ð1 þ sÞ.

Step 4. For n¼ 0; 1, compute Wnusing(23).

It follows from(23)that W0¼ 1 and W1¼ ð1 þ s þ 6s2þ 4s3þ s4Þ=ð1 þ sÞ.

Step 5. For n¼ 0; 1, compute KðnÞ using(28).

It finds from(28)that Kð0Þ ¼ 9lð1  cÞ 2 c2 and Kð1Þ ¼ 3 lð1  cÞð3  2cÞð1 þ sÞ2 c2 :

Step 6. For n¼ 0; 1, compute P1;nð0Þ using(27)in terms of P0,0.

Using(27), we finally get

P1;0ð0Þ ¼ ½W1Kð1Þ þ W0Kð0ÞP0;0¼ 3lð1  cÞð3  2cÞð1 þ sÞ2ð1 þ s þ 6s2_{þ 4s}3_{þ s}4_{Þ  9lð1  cÞ}2 c2 P0;0; P1;1ð0Þ ¼ W0Kð1ÞP0;0¼ 3lð1  cÞð3  2cÞð1 þ sÞ3 c2 P0;0:

Step 7. For n¼ 1; 2, compute P

1;nð0Þ using(29)in terms of P0,0.

It implies from(29)that

P_1;1ð0Þ ¼3ð1  cÞð3  2cÞsð3 þ 9s þ 17s 2_{þ 15s}3_{þ 6s}4_{þ s}5_Þ c2 P0;0; P_1;2ð0Þ ¼3ð1  cÞð3  2cÞsð3 þ 3s þ s 2_Þ c2 P0;0:

Step 8. Compute P_1;0ð0Þ using(32)in terms of P0,0.

Using(32)yields P_1;0ð0Þ ¼ 1 c2f3sð1  cÞ½cð3  12s  36s 2_{ 78s}3_{ 78s}4_{ 34s}5_{ 6s}6_Þ þ sð18 þ 54s þ 117s2_{þ 117s}3_{þ 51s}4_{þ 9s}5_ÞgP 0;0:

Step 9. Determine P0,0using(33). Thus P0;nð0Þ ðn ¼ 1; 2; 3Þ are achieved from Step 2, and P1;nð0Þ ðn ¼ 0; 1; 2Þ

are achieved from Step 7 and Step 8.

P0;0¼ c2½18  cð27  10cÞ þ 27sð2 þ 6s þ 12s2þ 18s3þ 15s4þ 6s5þ s6Þ  9scð27 þ 30s þ 60s2þ 270s3

þ 75s4_{þ 30s}5_{þ 5s}6_{Þ þ 9sc}2_{ð3 þ 12s þ 24s}2_{þ 36s}3_{þ 30s}4_{þ 12s}5_{þ 2s}6_Þ1_:

Example 3 (For D/M/1/K queueing system). We set the mean interarrival time b1 ¼ 1=k, F ¼ 1, and K ¼ 3,

where k is the interarrival rate. In this case, we have aðhÞ ¼ eh=k_:

Step 1. For each n¼ 0; 1; 2; 3, compute /nusing(16).

Using(16), we obtain

/0¼ 1; /1¼ ð1  aÞ=a; and /2¼ /3¼ ð1  aÞ=a 2_;

where a¼ l=ðl þ bÞ:

Step 2. For each n¼ 1; 2; 3, compute P

0;nð0Þ using(14)in terms of P0,0.

Using(14), we finally get P_0;1ð0Þ ¼ /1P0;0¼ 1 a a P0;0; P  0;2ð0Þ ¼ P  0;3ð0Þ ¼ /2P0;0¼ 1 a a2 P0;0:

Step 3. Compute ‘1and j1using(22) and (25), respectively.

Using(22)yields ‘1¼ r, where r ¼ l=k. Form(25), we obtain j1¼ er r.

Step 4. For each n¼ 0; 1, compute Wn using(23).

It implies from(23)that W0¼ 1 and W1¼ er r.

Step 5. For each n¼ 0; 1, compute KðnÞ using(28).

It follows from(28)that Kð0Þ ¼ lð1aÞ_a2 2and Kð1Þ ¼

lð1aÞer

a2 .

Step 6. For each n¼ 0; 1, compute P1;nð0Þ using(27)in terms of P0,0.

Using(27)yields P1;0ð0Þ ¼ ½W1Kð1Þ þ W0Kð0ÞP0;0¼ lð1  aÞðe2r_{ re}r_{þ a  1Þ} a2 P0;0; P1;1ð0Þ ¼ W0Kð1ÞP0;0 ¼ lð1  aÞer a2 P0;0:

Step 7. For each n¼ 1; 2, compute P

1;nð0Þ using(29)in terms of P0,0.

It implies from(29)that P_1;1ð0Þ ¼ð1  aÞðe 2r_{ re}r_{ 1Þ} a2 P0;0 and P  1;2ð0Þ ¼ ð1  aÞðer_{ 1Þ} a2 P0;0: Step 8. Compute P 1;0ð0Þ using(32)in terms of P0,0. Using(32)yields P1;0ð0Þ ¼ ð1aÞ½2þarð1rÞer_ðer_þ1rÞ a2 P0;0.

Step 9. Determine P0,0using(33). Thus P0;nð0Þ ðn ¼ 1; 2; 3Þ are achieved from Step 2, and P1;nð0Þðn ¼ 0; 1; 2Þ

are achieved from Steps 7 to 8. P0;0¼

a2

re2r_{ð1  aÞ þ re}r_{ð1  aÞð1  rÞ þ 2  a þ ar  a}2_r:

4. Optimal F-policy

Some important system performance measures of the F-policy G/M/1/K queueing system with exponential startup time are first defined as follows:

Ls the expected number of customers in the system;

Pb the probability that the server is busy;

Ps the probability that the server requires a startup time before starting the service;

Pbl the probability that the server is blocked.

The expressions for Ls, Pb, Ps, and Pbl are given by

Ls¼ XK n¼1 nP0;nþ XK1 n¼1 nP1;n; Pb¼ XK n¼0 P0;nþ X K1 n¼0 P1;n;

Ps¼ XF n¼0 P0;n; Pbl¼ XK n¼0 P0;n:

Next, we develop the total expected cost function per unit time for the F-policy G/M/1/K queueing system with startup times, in which F is a decision variable. The main purpose of this paper is to determine the opti-mum operating F-policy so as to minimize this total expected cost function. Let

Ch holding cost per unit time for each customer present in the system;

Cb cost per unit time for a busy server;

Cs startup cost per unit time for the preparatory work of the server before starting the service;

Cbl fixed cost for every lost customer when the system is blocked.

Utilizing the definitions of each cost element listed above, the total expected cost function per unit time is given by

TCðF Þ ¼ ChLsþ CbPbþ CsPsþ CblkPbl: ð34Þ

The optimal value of F, F*_{is determined by the following inequalities:}

TCðF 1Þ P TCðFÞ and TCðFþ 1Þ P TCðFÞ: ð35Þ

5. Numerical comparisons

We set the system capacity K¼ 15. We perform a sensitivity analysis for changes in the optimum value F*

along with changes in specific values of the system parameters. We consider three simple examples for three different interarrival time distributions such as exponential, 3-stage Erlang, and deterministic. The following cost elements are employed:

Case 1: Ch= 10, Cb= 200, Cs= 250, Cbl= 350. Case 2: Ch= 10, Cb= 200, Cs= 250, Cbl= 400. Case 3: Ch= 10, Cb= 200, Cs= 300, Cbl= 400. Case 4: Ch= 10, Cb= 225, Cs= 300, Cbl= 400. Case 5: Ch= 15, Cb= 225, Cs= 300, Cbl= 400. Table 1

The optimal value of F and its minimum expected cost for exponential interarrival time

ðl; bÞ ¼ ð1:0; 3:0Þ ðk; bÞ ¼ ð0:7; 3:0Þ ðk; lÞ ¼ ð0:7; 1:0Þ k l b 0.55 0.65 0.75 1.0 1.1 1.2 2.0 4.0 5.0 Case 1 F* _{6 4 3 4 6 8 4 4 4} TC(F*_{) 122.209 148.361 177.914 162.635 144.660 130.652 162.654 162.626 162.620} Case 2 F* _{8 6 5 5 8 11 5 5 5} TC(F*_{) 122.215 148.425 178.303 162.803 144.705 130.663 162.823 162.793 162.787} Case 3 F* _{8 6 5 5 8 11 5 5 5} TC(F*_{) 122.216 148.428 178.318 162.810 144.708 130.664 162.834 162.798 162.791} Case 4 F* _{7 5 4 4 7 10 5 4 4} TC(F*_{) 135.963 164.647 196.879 180.230 160.598 145.243 180.253 180.218 180.211} Case 5 F* _{4 3 2 2 4 6 2 2 2} TC(F*_{) 142.056 173.713 210.174 191.254 169.196 152.207 191.276 191.242 191.235}

In this section we provide the numerical results of the optimal value F*_{and the minimum expected cost for}

three interarrival time distributions and specific values of k, l, b. We first fixðl; bÞ ¼ ð1:0; 3:0Þ and choose

different values of k¼ 0:55; 0:65; 0:75. Next, we fix ðk; bÞ ¼ ð0:7; 3:0Þ and consider various values of

l¼ 1:0; 1:1; 1:2. Finally, we fix ðk; lÞ ¼ ð0:7; 1:0Þ and select different values of b ¼ 2:0; 4:0; 5:0.

The optimal value of F, F*_{, and its minimum expected cost TCðF}_{Þ for the above five cases are shown in}

Tables 1–3. For fixed values of ðl; bÞ and various values of k in Tables 1–3, we observe that (i) TCðFÞ

increases as k increases for any case; and (ii) F* _{decreases as k increases for any case. For fixed values of}

ðk; bÞ and various values of l inTables 1–3, we find that (i) TCðF_{Þ decreases as l increases for any case;}

and (ii) F*_{increases as l increases for any case. Again, for fixed ðk; lÞ and various values of b in Tables}

1–3, we observe that (i) TCðF_{Þ slightly decreases as b increases for any case; and (ii) F}*_{does not change}

at all when b changes from 2.0 to 5.0 for any case. Intuitively, F*_{is insensitive to changes in b.}

It can be easily seen fromTables 1–3that (i) F*_{increases as C}

hdecreases (see cases 4–5); and (ii) Chhas a

larger effect on F*_{than C}

b, Csand Cbl(see cases 3–4, cases 2–3 and cases 1–2).

6. Conclusions

The analytical steady-state results developed in this paper would be useful, which is significant to practitio-ners and system desigpractitio-ners. The main objective of this paper is threefold. We have first provided a recursive

Table 2

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

ðl; bÞ ¼ ð1:0; 3:0Þ ðk; bÞ ¼ ð0:7; 3:0Þ ðk; lÞ ¼ ð0:7; 1:0Þ k l b 0.55 0.65 0.75 1.0 1.1 1.2 2.0 4.0 5.0 Case 1 F* _{7 5 4 4 7 9 4 4 4} TC(F*_{) 118.991 143.288 170.687 156.448 139.842 126.868 156.450 156.447 156.447} Case 2 F* _{9 7 5 6 9 12 6 6 6} TC(F*_{) 118.991 143.291 170.747 156.463 139.844 126.868 156.465 156.462 156.462} Case 3 F* _{9 7 5 6 9 11 6 6 6} TC(F*_{) 118.991 143.291 170.750 156.464 139.844 126.868 156.466 156.463 156.462} Case 4 F* _{8 6 4 5 8 11 5 5 5} TC(F*_{) 132.741 159.539 189.471 173.957 155.752 141.451 173.959 173.956 173.955} Case 5 F* _{5 4 3 3 5 7 3 3 3} TC(F*_{) 137.236 166.178 199.711 182.155 162.033 146.552 182.157 182.153 182.153} Table 3

The optimal value of F and its minimum expected cost for deterministic interarrival time

ðl; bÞ ¼ ð1:0; 3:0Þ ðk; bÞ ¼ ð0:7; 3:0Þ ðk; lÞ ¼ ð0:7; 1:0Þ k l b 0.55 0.65 0.75 1.0 1.1 1.2 2.0 4.0 5.0 Case 1 F* _{10 6 4 5 7 10 5 5 5} TC(F*_{) 117.440 140.710 166.469 153.130 137.435 125.030 153.130 153.129 153.129} Case 2 F* _{10 7 5 6 9 12 6 6 6} TC(F*_{) 117.440 140.710 166.477 153.131 137.435 125.030 153.131 153.130 153.130} Case 3 F* _{12 7 5 6 9 12 6 6 6} TC(F*_{) 117.440 140.710 166.478 153.131 137.435 125.030 153.131 153.131 153.130} Case 4 F* _{9 6 5 5 8 11 6 5 5} TC(F*_{) 131.190 156.960 185.224 170.630 153.344 139.613 170.630 170.630 170.630} Case 5 F* _{6 4 3 4 6 8 4 4 4} TC(F*_{) 134.911 162.315 193.445 177.193 158.425 143.794 177.193 177.193 177.193}

method for obtaining the steady-state probability distributions of the number of customers in the system. Next, we have illustrated our recursive method by a study of three different interarrival time distributions: exponential, 3-stage Erlang, and deterministic. In addition, we provide a very efficient solution algorithm

to calculate the optimal threshold F* _{at minimum cost. Finally, we have performed a sensitivity analysis}

among the optimal value of F, specific values of system parameters, and the cost elements. Further, the devel-oped controlling arrival systems in this paper can be modeled many quality and service (Q&S) system in real-life.

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