Sensitivity analysis of the optimal management policy for a queuing system with a removable and non-reliable server

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Sensitivity analysis of the optimal management policy

for a queuing system with a removable and non-reliable server

q

Wen Lea Pearn

a

, Jau-Chuan Ke

b,

*, Ying Chung Chang

c

a

Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, ROC

b

Department of Statistics, National Taichung Institute of Technology, Taiwan, ROC

c_{Department of Industrial Engineering and Management, Ching Yun University, Taiwan, ROC}

Accepted 10 November 2003

Abstract

The management policy of an M/G/1 queue with a single removable and non-reliable server is considered. The decision-maker can turn the single server on at any arrival epoch or off at any service completion. It is assumed that the server breaks down according to a Poisson process and the repair time has a general distribution. Arrivals form a Poisson process and service times are generally distributed. In this paper, we consider a practical problem applying such a model. We use the analytic results of the queueing model and apply an efficient Matlab program to calculate the optimal threshold of management policy and some system characteristics. Analytical results for sensitivity analysis are obtained. We carry out extensive numerical computations for illustration purposes. An application example is presented to display how the Matlab program could be used. The research is useful to the analyst for making reliable decisions to manage the referred queueing system.

Keywords: Analytical results; Management policy; Non-reliable; Sensitivity analysis

1. Introduction

In this paper we study the operational characteristics of an M/G/1 queueing system in which a removable and non-reliable server operates with an N policy. The term ‘removable server’ is just an abbreviation for the system of turning on and turning off the server, depending on the number of

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q

This manuscript was processed by Area Editor Gary L. Hogg. * Corresponding author.

E-mail addresses: jauchuan@mail.ntit.edu.tw (J.-C. Ke); roller@cc.nctu.edu.tw (W.L. Pearn); ycchang@cyu.edu.tw (Y.C. Chang).

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customers in the system. A non-reliable server means that the server is typically subject to unpredictable breakdowns. The server is removable and applies the N policy: turn the server on whenever N ðN $ 1Þ or more customers are present, turn the server off only when no customers are present. After the server is turned off, the server may not operate until N customers are present in the system.

For a reliable server, the N policy M/M/1 queueing system was first developed by Yadin and Naor (1963), and the N policy M/G/1 queueing system was developed be several researchers such as Bell (1971, 1972), Heyman (1968), Kimura (1981), Teghem (1987), Tijms (1986), Artalejo (1998), and Wang and Ke (2000). For a non-reliable server,Avi-Itzhak and Naor (1963)studied the ordinary M/M/ 1 queueing system where the service rule does not depend on the number of customers in the queue. The ordinary M/Ek/1 queueing system with arrival rate depending on server breakdowns,

was investigated by Shogan (1979). Neuts and Lucanton (1979) studied a Markovian queueing system with multiple servers subject to breakdowns and repairs. The explicit solutions for the N policy Markovian queueing systems with a non-reliable server may be used to obtain the results for the N policy M/M/1 queueing system with a reliable server (seeSivazlian & Stanfel, 1975), or the ordinary M/M/1 queueing system with a non-reliable server (seeWang, 1990), or the ordinary M/M/1 queueing system with a reliable server (see Sivazlian & Stanfel, 1975) as a special case.

The purpose of this paper is threefold. First, an efficient Matlab program is used to calculate the optimal policy value N and some system characteristics. Second, the analytical results of the sensitivity analysis are derived. We then carry out extensive numerical computation for sensitivity analysis purpose. Third, we present an application example showing the way in which the Matlab program is used to calculate system characteristics, the optimum value of N and its minimum expected cost for various system parameters, while maintaining the maximum service quality.

Note that existing research works for the queueing system have never investigated the analytic solutions for the sensitivity analysis. In this paper, we will completely and successfully perform the sensitivity analysis for the M/G/1 queueing system with a removable and non-reliable server. Through this sensitivity analysis, we will be able to analyze the complex but exact solutions for a practical and general queueing system.

2. The queueing service model

2.1. System description and assumptions

Referring to Wang and Ke (2002), we consider the following model formulation. A cycle of the model consists of an idle period and a completion period. The completion period is composed into busy period and the breakdown period. As the system is empty, one cycle begins. The server is turned off until there are N customers in the system. We call this the idle period. The busy period is initiated when the server starts serving the waiting customers. While providing the service, the server may break down and be sent for repair immediately. This is called the breakdown period. As soon as the server is repaired, he returns to service again until all customers in the system are serviced. Since the completion period starts when the idle period is over and terminates when there are no customers in the system, the completion period may be represented as the sum of the busy period and the breakdown period. In addition, we consider the model under the following assumptions:

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(1) the arrival process is a Poisson process with rate l: Arriving customers at the server form a single waiting line and are serviced in the order of their arrivals (FCFS). The server may serve only one customer at a time. The service times constitute a set of set of independent and identically distributed random variables with a common distribution function FSðtÞ; a mean mS and a finite variance s2S: (2) Whenever the system is empty the idle period starts. When the server finds at least N customers waiting in the system, he begins the service immediately until the system is empty again. (3) When the server is working, the server may break down at any time with a Poisson breakdown rate a: (4) When the server fails, it is immediately repaired in a repair facility, where the repair times are independent and identically distributed random variables with a common distribution function FRðtÞ; a mean mR and a finite variance s2R: (5) If the server fails or a customer is in service, then newly arriving customers or waiting customers have to wait in the queue until the server is free. The service is interrupted if the server breaks down, and the server is immediately sent for repair. When the repair is completed, the server immediately returns for service.

2.2. Steady-state results

2.2.1. The expected number of customers in the system

We define the expected number of customers in the system for the M/G/1 queueing system under the N policy with server breakdowns as E½Ns: From Wang and Ke (2002) we have the following expression. E½Ns ¼ ðN 2 1Þ 2 þ rð1 þ amRÞ þ l2½ð1 þ amRÞ2ðm2Sþ s2SÞ þ amSðm2Rþ s2RÞ 2½1 2 rð1 þ amRÞ ; ð1Þ where r ¼ lmS:

2.2.2. Other system characteristics

Define E½I ; the expected length of the idle period; E½B ; the expected length of the busy period; E½D ; the expected length of the breakdown period; E½C ; the expected length of a cycle. Using the results stated inWang and Ke (2002), we have the long-run fraction of time, that the server is idle, busy, or broken down, respectively, are:

E½I E½C ¼ 1 2 rð1 þ amRÞ; ð2Þ E½B E½C ¼ r; ð3Þ E½D E½C ¼ almSmR; ð4Þ

and the number of cycles per unit time is

1 E½C ¼

l½1 2 rð1 þ amRÞ

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3. Optimal management policy

3.1. Total expected cost function

We develop the total expected cost function per unit time for the N policy M/G/1 queueing system with a non-reliable server, in which N is a decision variable. With the cost structure being constructed, our objective is to determine the optimal management policy to minimize this cost function, while maintaining the maximum service quality to the customers. Let Ch; holding cost per unit time for each customer present in the system; Co; cost per unit time for keeping the server on; Cf ; cost per unit time for keeping the server off; Cb; breakdown cost per unit time for a broken server; Cs ; start-up cost for turning the server on. Based on the definitions of each cost element, and its corresponding system characteristics, the total expected cost function per unit time is given by

FðNÞ ¼ ChE½Ns þ Co E½B E½C þ Cf E½I E½C þ Cb E½D E½C þ Cs 1 E½C: ð6Þ

It is to be noted that the second and third terms of Eq. (1) are independent of the decision variable N: Likewise, we note from Eqs. (2) – (4) that, terms E½B=E½C; E½I=E½C; and E½D=E½C do not involve the decision variable N: Discarding those cost terms that are not a function of the decision variable N; the optimization problem reduces to

minimize ~FðNÞ ¼ Ch N 2 1

2 þ Cs

N : ð7Þ

Omitting the fixed cost—(1/2) Ch of the first term, Eq. (7) reduces to minimize ^FðNÞ ¼ Ch N 2 þ Cs l½1 2 rð1 þ amRÞ N ; ð8Þ subject to 0 , r , 1; and N ¼ 1; 2; …:

3.2. Determining the optimal management policy

Since N is a positive integer, N ¼ 1; 2,…, the optimal value of N; Np; to minimize FðNÞ is determined by satisfying the following inequalities

^

FðNp2 1Þ $ ^FðNpÞ; ^

FðNpþ 1Þ $ ^FðNpÞ: ð9Þ

From Eq. (8), the necessary conditions for Npto be optimal reduces to

ðNp2 1ÞNp# 2lCs½1 2 rð1 þ amRÞ Ch

# NpðNpþ 1Þ: ð10Þ

Note that a double solution is possible.

Differentiating FðNÞ with respect to N and setting the result equal to zero yields Ch

2 2 Cs

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Thus, the optimal value of N is approximately given by Np¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2lCs½1 2 rð1 þ amRÞ Ch s : ð11Þ

Differentiate FðNÞ with respect to N twice and then substitute N ¼ Npto obtain

d2FðNpÞ

dN2 ¼

2lCs½1 2 rð1 þ amRÞ

N3 . 0; ð12Þ

which implies that FðNÞ is a concave upward (convex) function and achieves a global minimum when N ¼ Np: We note that if Npis not an integer, the optimum value of N is one of the integers closest to Np; the expression may rewrite as

Np¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2lCs½1 2 rð1 þ amRÞ Ch s þ 1; ð13Þ where 1 [ ð21; 1Þ is a constant.

For the special case, N-policy M/M/1 queueing system with a removable and non-reliable server, where the service and repair times are assumed to follow the exponential distribution with means mS and mR; respectively (see Wang, 1995), with some similar algebraic manipulations. It is interesting to note that we would obtain the same expression for the optimal value Np as stated in Eq. (13).

4. Analytical results for sensitivity analysis

We now perform a sensitivity analysis on the optimum value Np based on changes in considerable values of the cost parameters Ch; Co; Cf; Cb; Cs and system parameters l; mS; a; and mR: It is to be noted that the terms E½B=E½C; E½I=E½C; and E½D=E½C do not involve the decision variable N: We may set the relative cost parameters Co; Cf; and Cb to be some fixed constants. Eq. (13) suggests that Np/ ffiffiffiffiffiffiffiCs=Ch

p

; which is straightforward. Differentiate Np with respect to l to obtain ›Np ›l ¼ ffiffiffiffiffiffiffi Cs=Ch p ½1 2 2rð1 þ gÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2l½1 2 rð1 þ gÞ p _; ð14Þ where g ¼ amR:

On putting this last expression equal to 0 and solving for l; we find l ¼ 1=mS½2ð1 þ gÞ: (Note that l , 1=mS½ð1 þ gÞ is required.) Differentiate ›N

p

=›l with respect to l again and substitute l ¼ 1=mS½2ð1 þ gÞ to get ›2Np ›l2 l¼1=mS½2ð1þgÞ ¼ 22 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Csm3Sð1 þ gÞ 3 Ch s , 0; ð15Þ

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which implies Np is a concave downward function with respect to l and obtains its maximum at l ¼ 1=mS½2ð1 þ gÞ: Differentiating N

p

with respect to mS; we have ›Np ›mS ¼ 2ð1 þ gÞl 2 ffiffiffi_C s p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2lCh½1 2 rð1 þ gÞ p , 0; ð16Þ

for l , 1=mS½ð1 þ gÞ; ;mS: This means that N p decreases in mS: Differentiate N p with respect to a to find ›Np ›a ¼ 2rpffiffiffiffiffilC_s mR ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Ch½1 2 rð1 þ gÞ p , 0; ð17Þ

for l , 1=mS½ð1 þ gÞ; ;a: This means that N p

decreases in a: Differentiate Npwith respect to mRwe get ›Np ›mR ¼ 2lra ffiffiffi Cs p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Ch½1 2 rð1 þ gÞ p , 0; ð18Þ

for l , 1=mS½ð1 þ gÞ; ;mR: This means that N p

decreases in mR: We have the following analytic sensitivity analysis results:

1. Npincreases in l for r , 1=½2ð1 þ gÞ and decreases in l for r . 1=½2ð1 þ gÞ:

2. Npdecreases in mS: For the queueing system with service times described by an arbitrary probability distribution with mean mS; it is to be noted that only the mean service time would affect the sensitivity results.

3. Npdecreases in a for large r and g (for rð1 þ gÞ closes to 1), whereas it is insensitive to a for small r and g ðrð1 þ gÞ p 1Þ.

4. Npdecreases in mRfor large r and g (for rð1 þ gÞ closes to 1), whereas it is insensitive to mRfor small r and small g (for rð1 þ gÞ p 1).

5. Co; Cf; and Cb do not affect N p

: 6. Npis proportional to ffiffiffiffiffiffiffiCs=Ch

p

: In other words, Np

increases in Cs whereas it decreases in Ch: It is noted that the results of the sensitivity investigation in Wang (1995) were not correctly interpreted.

5. Numerical computations

5.1. Parameters setting

We now perform a numerical illustration of the sensitivity analysis on the optimum value Npbased on changes in considerable values of system parameters l; mS; a; and mR: It is to be noted that the terms E½B=E½C; E½I=E½C; and E½D=E½C do not involve the decision variable N: We may set the relative cost parameters Co; Cf; and Cb to be some fixed constants. Additionally, incremental, rather than accounting costs are considered, since the latter often include such nonincremental elements as overhead. Eq. (13) suggests that Np/ ffiffiffiffiffiffiffiCs=Ch

p

: And we note that the N policy is applied to control the queueing system due to expensive start-up cost per cycle (relative to holding cost). In purpose to

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demonstrate some of the sensitivity analysis results, Cs¼ $1600; $4000 have been set to represent two levels of cost relationship and Chis selected to be $5.

The sensitivity calculations demonstration may now focus on the four critical input parameters: l; mS; a and mR: We group them into two pairs: l and mS; a and mR; under consideration simultaneously in order to study the interaction of the key factors. Individual affection on the optimal solution is examined as well. We now consider the following experimental design of system parameters for sensitivity analysis on the optimum value Npbased on changes in considerable input values. Note that 0 , r , 1 and 0 , g , 1 are sufficient for stationary. We calculate the optimal value Np for the parameters settings summarized in Table 1, which cover a widespread range of applications dealing with the referred queueing model.

Rows 2 – 4 list the parameters settings for various combinations of l and mS: The specified range l ¼ 0:05 (0.0125) 0.95 and m21_S ¼ 1 (0.125) 10 is considered in which r ¼ lmS covers widespread traffic intensities in real application. Rows 3 and 4 are chosen to examine the sensitivity of Npversus l or mSonce at a time. In row 3, l ¼ 0:05 (0.0125) 0.95 and three levels of m21S ¼ 1; 1.5, and 3 are selected. In row 4, three levels of l ¼ 0:35; 0.65, 0.95 and m21S ¼ 1ð0:125Þ10 are considered. Rows 5–7 list the parameters settings for various combinations of a and mR: The range a ¼ 0:01 (0.018) 0.91 and m21R ¼ 1ð0:18Þ10 are considered in which g ¼ amRcovers widespread real applications. In rows 6 and 7, the sensitivity of Npversus a or mRare examined once at a time. Row 6 demonstrates the relationship of Npversus a for three levels m21R ¼ 1; 1.9, 3.7. Row 7 illustrates the relationship of N

p

versus mRfor three levels a ¼ 0:28; 0.64, 0.91.

Figures are arranged in the following way:Fig. 1(a)plots the surface of Npversus l ¼ 0:05 (0.0125) 0.95 and m21S ¼ 1 (0.125) 10 for various parameters settings given inTable 1.Fig. 1(b) and (c)are the cross-section, which plot the curves of Npversus l and mS; respectively. Three levels of the other system parameter are picked (seeTable 1).Fig. 2(a)plots the surface of Npversus a ¼ 0:01 (0.018) 0.91 and m21_R ¼ 1ð0:18Þ10 for various parameters settings given inTable 1.Fig. 2(b) and (c)are the cross-section plot the curves of Np versus a and mR; respectively. Three levels of the other system parameter are picked as listed inTable 1.

5.2. Interpretation of the results in figures

Fig. 1(a) – (c) reveal that: (i) Np increases in l for l , 1=mS½2ð1 þ gÞ and decreases in l for l . 1=mS½2ð1 þ gÞ: The ‘local maximum’ 1=mS½2ð1 þ gÞ is moving from left to right as mS

Table 1

Parameters settings for various system parameters combinations with fixed costs Co¼ 100; Cf¼ 20; and Cb¼ 200

Ch Cc l m21S a m21R Parameter setting (1) Parameter setting (2)

5 1600 (1) (2) 0.2 1 0.05(0.0125)0.95 1(0.125)10 5 1600 (1) (2) 0.2 1 0.05(0.0125)0.95 1, 1.5, 3 5 1600 (1) (2) 0.2 1 0.35, 0.65, 0.95 1(0.125)10 5 4000 0.3 1 (1) (2) 0.01(0.018)0.91 1(0.18)10 5 4000 0.3 1 (1) (2) 0.01(0.018)0.91 1, 1.9, 3.7 5 4000 0.3 1 (1) (2) 0.28, 0.64, 0.91 1(0.18)10

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decreases. If mSis small enough, one could see that N p

increases in l (seeFig. 1(b)). (ii) Npdecreases in mS (seeFig. 1(c)). One can see fromFig. 1(a)that N

p

increases in l but decreases in mS for small mS: We observe fromFig. 2(a) and (c)that: (i) Npdecreases in a for large g; whereas it is insensitive to a for small g (seeFig. 2(a) and (b)). (ii) Npdecreases in mRfor large g (seeFig. 2(a) and (c)), whereas it is insensitive to mR for small g (see Fig. 2(a) and (c)). It is interesting to see that for a wide range of combinations of a and mRinFig. 2(a), N

p

is insensitivity to both parameters.

6. An application example

M/G/1 queueing systems arise naturally as models for many computer communication systems, production systems and integrated manufacturing systems. This section gives an example of the way in which the Matlab program can be used by an analyst to calculate system characteristics, the optimum value of N and its minimum expected cost. The application illustrates the levels of detail that are

Fig. 1. (a). Surface plot of Np versus l ¼ 0:05ð0:0125Þ0:95 and m21S ¼ 1ð0:125Þ10: (b). Plots of N p

versus l ¼ 0:05ð0:0125Þ0:95 for m21

S ¼ 1; 1.5, 3 (bottom to top in plot). (c). Plots of N p

versus m21

S ¼ 1ð0:125Þ10

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appropriate for building a model and using that model for performance projection. The example illustrates the relationship between modeling concepts, evaluation algorithms, and modeling software. The example also indicates how such software can save the cost by the analyst.

Consider the following example. Computer communication networks use a variety of flow control policies to achieve high throughput, low delay, and stability. Here, we model the flow control policy of IBM’s System Network Architecture (SNA). SNA routes messages from sources to destinations by way of intermediate nodes which temporarily buffer the messages. Messages buffers are a scarce resource. The flow control policy regulates the flow of messages between source/destination pairs in an effort to avoid problems such as deadlock and starvation, which could result from poor buffer management.

SNA has a window flow control policy. The key control parameter is the window size, N: There is a ‘message generation center’ (server) and a ‘pacing box’. Together, the message generation center and the pacing box mimic the flow control policy, in the following way. When a source starts sending messages to a particular destination, a pacing count at the source is initialized to the value of zero. This pacing count is incremented every time a message is received. The pacing box ‘stores’ up to N 2 1 messages. When the Nth message arrives, it triggers the discharge of all

Fig. 2. (a). Surface plot of Npversus a ¼ 0:01ð0:018Þ0:91 and m21

R ¼ 1ð0:18Þ10: (b). Plots of N p

versus a ¼ 0:01ð0:018Þ0:91 for m21R ¼ 1; 1.9, 3.7 (bottom to top in plot). (c). Plots of N

p

versus m21R ¼ 1ð0:18Þ10 for a ¼ 0:28; 0:64; 091 (top to bottom

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N messages into the queue of the message generation center to be served. The service time of the message generation center has a general distribution with mean mS; as long as its queue is non-empty, it will generate message traffic at this rate. One assumption made by this modeling approach is that ‘message generation center’ is subject to breakdown at any time while it is working. Whenever the ‘message generation center’ fails, it is immediately repaired. After the repair the interrupted service is resumed. Breakdowns occur only when the service is in progress, and never when the system is empty. As soon as the server (message generation center) is repaired, it functions as good as a new one. Our objective is to model the ‘pacing level’ of messages between a single source/destination pair-the optimum window size N to minimize the total expected cost. Customers, which represent messages, arrive at the source node at rate l: They flow node to node, requiring the message generation center to service at a mean time mS: A single coaxial cable is used to interconnect stations. The source continues to transmit, regardless of the number of outstanding messages. This model is shown in Fig. 3.

The calculations for the model do not require complicated intermediate functions to be implemented, and most of the system characteristics usually of interest can be calculated in a straightforward way. In the example investigated, input system parameters the message stream arrival rate l ¼ 0:4 message/s, the mean time of message generating (service) mS¼ 1:0 s/message, the standard deviation of message generating (service) sS ¼ 1:0 s/message, the breakdown rate a ¼ 0:05 unit/s, the mean time to repair mR¼ 0:2 s/unit, the standard deviation of repair sR ¼ 1:0 s/message, and cost element the holding cost per second for each message present in the system set to Ch¼ $5; the cost per second for keeping the ‘message generation center’ (server) working set to Co¼ $50; the cost per second for keeping the ‘message generation center’ off set to Cf¼ $10; the breakdown cost per second for a broken ‘message generation center’ set to Cb¼ $100; and the start-up cost for turning the ‘message generation center’ on set to Cs ¼ $200:

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System parameters Notation Value

Message stream arrival rate l 0.4

Mean time of message generating (service) mS 1.0

Standard deviation of message generating (service) sS 1.0

Breakdown rate a 0.05

Mean time to repair mR 0.2

Standard deviation of message generating (service) sR 1.0

Holding cost per second for each message present in the system Ch 5

Cost per second for keeping the server working Co 50

Cost per second for keeping the server off Cf 10

Breakdown cost per second for a broken server Cb 100

Start-up cost for turning the server on Cs 200

The program output is shown in the following: The output is:

E½Ns ¼ 2:1848 E½B ¼ 6:7114 E½I ¼ 10 E½D ¼ 0:0671

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Np¼ 4 TcostðN

p

Þ ¼ 49:2042

The MATLAB computer program gives the expected number of messages in the system E½Ns ¼ 2:18 messages, the expected length of generating (busy) period E½B ¼ 6:71 s, the expected length of idle period E½I ¼ 10 second and the expected length of breakdown period E½D ¼ 0:07 s. The value of N for the optimal management policy, is Np¼ 4 units, and the corresponding minimum expected cost is found to be TcostðN

p

Þ ¼ $49:20: Fig. 4 plots the minimum expected cost TcostðNÞ versus N ¼ 1ð1Þ30: The plot shows that that the minimum expected cost indeed occurs when N ¼ 4:

System characteristics Notation Value

Expected number of messages in the system E½Ns 2.18

Expected length of busy period E½B 6.71

Expected length of idle period E½I 10

Expected length of breakdown period E½D 0.07

Optimal management policy Np 4

Minimum expected cost TcostðN

p

Þ 49.20

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