International Journal of Mathematical Archive-5(1), 2014, 125-133 Available online through www.ijma.info ISSN 2229 5046 MAXIMUM ENTROPY ANALYSIS OF UNRELIABLE SERVER M X /G/1 QUEUE WITH ESSENTIAL AND MULTIPLE OPTIONAL SERVICES Rachna Vashishtha Pandey* and Piyush Tripathi Amity University Uttar Pradesh, Malhour Campus, Lucknow- 226010, India. *Research Scholar at Bhagwant University, Ajmer, Rajasthan, India. (Received on: 25-11-13; Revised & Accepted on: 15-01-14) ABSTRACT In this paper we use the maximum entropy principle(mep) to find the approximate waiting time of an M X /G/1 model with k services; the first being essential (first essential service FES) whereas remaining k-1 of them as optional services (multiple optional service MOS). The server is subject to breakdown while rendering service to the customers. The customers arrive in the system in Poisson fashion in batches with arrival rate. After getting FES, the customer may opt for first of MOS, with some probability; after completing it, he may opt for next MOS with some other probability and so on. The service time of FES is generally distributed, while service times of MOS are exponentially distributed. The noble feature of the present study is to employ MEP which helps us in finding precise performance measures. The derived approximate results based on MEP, are compared with exact results obtained in previous studies for different distributions as special cases. It is noticed that MEP provides an alternative approach for solving complex queueing systems, in particular when queue size distribution is to be computed. Keywords: Unreliable server, Bulk queue, Essential service, Multiple Optional services, Maximum entropy, Waiting time. 1. INTRODUCTION The waiting time of M X /G/1 queue with first essential service and multiple optional services where the server is unreliable is determined, using maximum entropy principle. In recent past, maximum entropy principle (MEP) has been used in various forms by several researchers Bard, 1980; Guiasu, 1986; Kouvatsos, 1989. The maximum entropy principle has been widely applied to the study of more complicated queueing systems having general inter-arrival times or general service times. Shore, 1978; utilized the MEP to develop some system performance measures for M/M//N, M/M/, and M/G/1 and G/G/1 models. El-Affendi and Kouvatsos, 1983; employed the maximum entropy principle to analyze the M/G/1 and G/M/1 queueing system. Kouvatsos, 1989; used MEP to derive the explicit expressions for the probability distribution of the queue length of G/G/1/N and G/G/C/N queueing systems. Wu and Chau, 1989; used principle of maximum entropy to analyze multiple- server queueing systems. Arizono et al. 1991 obtained the approximate steady solutions for M/M/R queueing system via ME approach. A single removable server in M/G/1 queueing system operating under N policy was studied by Wang et al, 2002. Jain and Jain, 2006; studied G/G/1 queue with vacation under N-policy using the principle of ME. Ke and Lin, 2007; used MEP for M X /G/1 queue, where the server operates under N policy and a single vacation. Recently, Maurya, 2013; in his paper, a bulk arrival retrial queueing model with two phases of service (M X /G 1, G 2 /1) has discussed under Bernoulli vacation schedule and explored mainly its steady state behavior using MEP. Much works on queueing models have been done for single service requirement however in real life service system, one encounters with numerous queueing situations where arriving customers demand not only main service but also some subsidiary service(s) which can be provided by the server. The study of two-phase service was proposed by Krishna and Lee, 1990. Selvam and Sivasankaran, 1994; considered a two-phase queueing system with server vacations. Madan, 1994, examined an M/G/1 queueing system with additional optional service and no waiting capacity. A single server Poisson input queue with a second optional channel was studied by Medhi, 2002. A two-stage service policy for M/G/1 queueing system where the speed of the server depends on the amount of work present in the system was studied by Lee and Kim, 2006.Choudhury and Tadj, 2011; explored an M X /G/1 Bernoulli vacation queue with two phases of service and unreliable server under N-policy. Corresponding author: Rachna Vashishtha Pandey* *Research Scholar at Bhagwant University, Ajmer, Rajasthan, India. E-mail: rachnav.pandey@yahoo.co.in International Journal of Mathematical Archive- 5(1), Jan. 2014 125
Due to wide applicability, the bulk queues have drawn the attention of many researchers working in the area of queueing theory. Choudhary and Templeton, 1983; in their book provided a comprehensive review on bulk queues and their applications. Madan et al., 2004; included the concept of an optional re-service for M X /G1, G 2 /1 queue. A twostage batch arrival queueing system with modified Bernoulli schedule vacation under N-policy was investigated by Choudhury and Madan, 2005. Choudhury, 2006; analyzed an M X /G/1 queueing system with two phases of heterogeneous service and Bernoulli vacation schedule that operate under classical retrial policy. A batch arrival queue with second optional service under N-policy has been studied by Choudhury and Paul, 2006. Ke, 2007; studied an M X /G/1 system with server s startup time and J additional options for service. It is an universal truth that the servers providing any kind of service are subject to fail due to some reason or the other. In many real life situations, due to overloading or long run operating time, the servers may break down. Queueing problems with server break down have been studied by many researchers cf. Avi-Itzhak and Naor, 1963. Wang, 2004 analyzed M/G/1 queue with second optional service and server breakdowns. Ke, 2006; investigated M/G/1 queueing system with unreliable server. Optimal control policy for M/G/1 queueing system with server breakdowns and general startup times has been studied by Wang et al. 2006. A batch arrival queue under vacation policies with server breakdowns and startup/closedown times was proposed by Ke, 2007. In the present investigation we employ MEP to analyze M X /G/1 queue with multi-optional service and server breakdowns. The rest of the paper is organized as follows. In section 2, stating requisite assumptions and notations we develop the maximum entropy results; the next section 3, contains the maximum entropy model, constructed on the base of some constraints in terms of long run probabilities. We derive explicit expressions for maximum entropy model in section 4. In section 5, we compare the exact and approximate waiting times. By means of numerical illustration, the sensitivity analysis is done in section 6. Finally we wind up our investigation with some concluding remarks in the last section 7. 2. NOTATIONS AND SOME RESULTS We use the maximum entropy principle for finding the steady state probabilities and mean waiting time of an unreliable server M X /G/1 queueing system with k services out of which the first service is an essential service (FES) whereas other services are optional services (MOS). After completion of FES if the customer demands for MOS, then the server may provide the MOS with probability r i ; i= 1, 2,..., k-1or becomes idle with probability (1-r i ); i=1,2,..., k-1. The ES is generally distributed but the MOS follow exponential distribution. The life time and repair time of the server during FES and MOS are generally distributed. The following notations are used to completely specify the queueing system: λ arrival rate µ l service rate for l th (l=1,2,,k) multiple optional service (MOS) X random variable denoting the batch size of customers C(z) probability generating functions (pgf) of the batch size X, i.e CC(zz) = 1 cc zz XX, XX 2 mean and 2 nd factorial moment of batch size B(.), b(.) cumulative distribution function (cdf) and probability density function (pdf) respectively of service time of FES R l (.), g l (.) cdf and pdf of repair time of l th (l=1,2,,k) MOS b(.), g l(.) Laplace Stieltjes transform (LST) of B(.) and R l (.), (l=2, 3,,k) ll failure rate of the server when it fails during the l th (l=1,2,,k) MOS γγ ll first moment of l th (l=1,2,,k) repair time Let ρρ ll = λλxx μμ ll aaaaaa δδ ll = 1 + αα ll γγ ll, ll = 1, 2,, Now ββ ll (xx) = bb(xx) 1 BB(xx) ; γγ ll(xx) = gg ll(xx) 1 RR ll (xx) ; ll = 1, 2,, The second moments of essential service time and (repair time are given as ββ ll (2) = ( 1) 2 bb (2) (0) aaaaaa γγ ll (2) = ( 1) 2 gg ll (2) (0); ll = 1,2,, For the steady state, let us define the following probabilities P 1 (n) n(>0) customers in the queue excluding the one being provided the FES and the server is busy P l (n) n(>0) customers in the queue excluding the one being provided the l th (l=2, 3, k) MOS and the server is busy 2014, IJMA. All Rights Reserved 126
Q 1 (n) Q l (n) n(>0) customers in the queue excluding the one being provided the FES and the server is being repaired n(>0) customers in the queue excluding the one being provided the l th (l=2, 3, k) MOS and the server is being repaired Some important analytical results are obtained. Theorem: 1 By introducing the supplementary variables X(t) and Y(t) corresponding to the elapsed service time for FES and repair time, respectively at time t can be constructed and solved. Some important analytical results are obtained i The long run probability of the server being busy when he provides the FES, is given by PP BB1 = PP 1 (nn) = ρρ 1 (1) ii The long run probability of the server being busy when he provides the l th (l=2, 3,,k) MOS, is given by PP BBBB = PP ll (nn) = ( nn =1 rr nn )ρρ ll, ll = 2,3,..., (2) iii The long run probability of the server is under repair when he provides the FES, is given by PP RR1 = QQ 1 (nn) = ρρ 1 αα 1 γγ 1 (3) iv The long run probability of the server being busy when he provides the l th (l=2, 3,,k) optional service, is given by PP RRRR = QQ ll (nn) = ( rr nn )ρρ ll αα ll γγ ll, ll = 2,3,, k (4) Theorem: 2 The average number of customers in the system is ) LL SS = ρρ 1 δδ 1 + rr 1 ρρ 2 δδ 2 + ll=3( rr nn ρρ ll δδ ll + EE(QQ) (5) Where EE(QQ) = 1 2 (λλxx ) 2 δδ 2 1 ββ (2) 1 + Δ 1 ββ 1 + rr 1 2ρρ 1 δδ 1 ρρ 2 δδ 2 + Δ 2 + 2(ρρ μμ 2 δδ 2 ) 2 2 + rr nn 2ρρ ll δδ ll ρρ nn δδ nn + Δ ll + 2(ρρ μμ ll δδ ll ) 2 1 1 ρρ 1 δδ 1 rr nn ρρ ll δδ ll ll ll=3 nn =1 and ii = λλxx 2 (1 + αα ii γγ ii ) + (λλxx ) 2 αα ii γγ ii (), ii = 1,2,..., 3. MAXIMUM ENTROPY PRINCIPLE Maximum entropy approach is a principle of reasoning that ensures that there is no unconscious arbitrary assumption about a system. Based on the principle we will propose a feasible method of numerical solutions for approximately analyzing a batch-arrival queueing system with essential service and multi- optional service. Due to Jaynes, 1957; the maximally noncommittal distribution with regard to missing information is that distribution which maximizes the entropy nn PP(nn) ln PP(nn) under the restrictions induced by the given information and which is least biased to the information missing. The entropy serves as a measure of the uncertainty of knowledge about the answer to a welldefined equation. 3.1. The maximum entropy model Using the principle of maximum entropy, we formulate the maximum entropy function for the M X /G/1 with FES and MOS as YY = PP 1 (nn) ln PP 1 (nn) ll=2 PP ll (nn) ln PP ll (nn) QQ 1 (nn) ln QQ 1 (nn) QQ ll (nn) ln QQ ll (nn) 2014, IJMA. All Rights Reserved 127 ll=2 ll=2 (6) The maximum entropy solution for our model is obtained by maximizing the above equations under the following constraints the probability that the server is providing FES:. PP 1 (nn) = ρρ 1 (7)
the probability that the server is providing MOS: PP ll (nn) = ( nn =1 rr nn )ρρ ll, ll = 2,3,..., (8) the probability that the server is broken during FES: QQ 1 (nn) = ρρ 1 αα 1 γγ 1 (9) the probability that the server is broken during MOS:. QQ ll (nn) = ( nn =1 rr nn )ρρ ll αα ll γγ ll, ll = 2,3,, k (10) the expected number of customers in the system: nn PP 1 (nn) + ll=2 PP ll (nn) + QQ 1 (nn) + ll=2 QQ ll (nn) = LL ss (11) where L s is obtained from equation (5). Multiplying each equation by θθ ii (i=1, 2, 2k+1), we obtain the Lagrangian function (y) as follows: yy = PP 1 (nn) ln PP 1 (nn) PP ll (nn) ln PP ll (nn) ll=2 QQ 1 (nn) ln QQ 1 (nn) QQ ll (nn) ln QQ ll (nn) θθ 1 PP 1 (nn) ρρ 1 rr nn ρρ ll θθ +1 ll=2 QQ 1 (nn) ρρ 1 αα 1 γγ 1 2 θθ ll ll=+2 nn =1 θθ ll ll=2 PP ll (nn) QQ ll (nn) rr nn ρρ ll αα ll γγ ll θθ 2+1 nn PP 1 (nn) + ll=2 PP ll (nn) + QQ 1 (nn) + ll=2 QQ ll (nn) LL ss (12) where θθ ii (i=1, 2, 2k+1) are the Lagrangian Multipliers corresponding to constriants (7) - (11), respectively. 3.2. The maximum entropy solutions To find the maximum entropy solution for P j (n) and Q j (n) (j=1,2,,k), we need to maximizing objective function given in equation (6), subject to the constraints(7)-(11); which is equivalent to maximizing objective function given by equation (12) only. The ME solutions are obtained by taking the partial derivatives of function y with respect to P j (n) and Q j (n) (j=1,2,,k) and setting the results equal to zero. Now PP jj (nn) = ln PP jj (nn) 1 θθ jj θθ 2+1 nn = 0, jj = 1,2,, (13) QQ jj (nn) = ln QQ jj (nn) 1 θθ +jj θθ 2+1 nn = 0, jj = 1,2,, (14) Solving the general equations (13) and (14), we obtain PP jj (nn) = ee (1+θθ jj +θθ 2+1 nn), jj = 1,2,, (15) QQ jj (nn) = ee (1+θθ +jj +θθ 2+1 nn), jj = 1,2,, (16) Let us denote φφ jj = ee 1+θθ jj ; φφ +jj = ee 1+θθ +jj, jj = 1,2,, ; φφ 2+1 = ee θθ 2+1 (17) PP jj (nn) = φφ jj φφ 2+1 nn, jj = 1,2,, (18) QQ jj (nn) = φφ +jj φφ 2+1 nn, jj = 1,2, (19) 2014, IJMA. All Rights Reserved 128
Substituting these values in equations (7)-(11), and after some algebraic manipulation, we get the following results: PP 1 (nn) = φφ 1 nn φφ 2+1 = φφ 1φφ 2+1 1 φφ 2+1 = ρρ 1 (20) Similarly, φφ llφφ 2+1 = ( rr 1 φφ nn =1 nn )ρρ ll, ll = 2,3,..., (21) 2+1 φφ +jj φφ 2+1 jj 1 = 1 φφ rr nn ρρ jj αα jj γγ jj, jj = 2,3,, k (22) 2+1 2 φφ nn φφ 2+1 (1 φφ 2+1 ) 2 = LL SS (23) Summing equations (20)-(22), we get 2 φφ nn φφ 2+1 nn =1 ) = ρρ (1 φφ 2+1 ) 1δδ 2 1 + ll=2( rr nn ρρ ll δδ ll = ρρ HH (24) Thus from equations (23) and (24), we have φφ 2+1 = LL SS ρρ HH LL SS (25) We now derive the values of φφ nn (n=1, 2,, 2k) as φφ 1 = ρρ 1ρρ HH ; φφ LL SS ρρ ll = ( nn =1 rr nn )ρρ ll ρρ HH, ll = 2, 3,, HH LL SS ρρ HH φφ +1 = ρρ 1αα 1 γγ 1 ρρ HH ; φφ LL SS ρρ +ll = nn =1 rr nn ρρ ll αα ll γγ ll ρρ HH, ll = 2, 3,, (26) HH LL SS ρρ HH From equations (17)-(18), we obtain PP 1 (nn) = ρρ 1ρρ HH (LL SS ρρ HH ) nn 1 (27) (LL SS ) nn PP ll (nn) = nn =1 rr nn ρρ ll ρρ HH (LL SS ρρ HH ) nn 1, ll = 2, 3,, (28) (LL SS ) nn QQ 1 (nn) = ρρ 1αα 1 γγ 1 ρρ HH (LL SS ρρ HH ) nn 1 (29) (LL SS ) nn QQ ll (nn) = nn =1 rr nn ρρ ll αα ll γγ ll ρρ HH (LL SS ρρ HH ) nn 1, ll = 2, 3,, (30) (LL SS ) nn 4. EXPECTED WAITING TIME IN THE QUEUE We shall now find the exact and approximate formulae for expected waiting time in the queue. The exact expected waiting time in the queue is obtained using Little s formula WW = EE(QQ) λλxx (31) In order to find the mean waiting time of a test customer at busy state and the repair state we proceed as follows. On arrival, the test customer finds n customers waiting in the queue and demanding for FES and MOS and the system is either in busy or repair state then I. In busy state: Since the server is working, the test customers wait till n customers get their FES and MOS. The mean waiting time of the test customer in the busy state is nnββ 1 for ES and nn for l th (l=2, 3,, k) OS. μμ ll II. In repair state: Let E(R i ) denote the mean residual repair time where i=1, 2,, k. Using the well known result of Kleinrock (1975), we have EE(RR ii ) = γγ ii 2,i=1, 2,, k. Thus the mean waiting time of the test customer when 2γγ ii the server is in repair state is nn + γγ ii 2 i=1, 2,,k.Now using the above results, we obtain the approximate μμ ll 2γγ ii expected waiting time in the queue, as 2014, IJMA. All Rights Reserved 129
WW qq = nn nnββ 1 PP 1 (nn) + ll=2 PP μμ ll (nn) + nnββ 1 + γγ 1 2 QQ ll γγ 1 (nn) + nn + γγ 1 2 ll=2 QQ 1 μμ ll γγ ll (nn) (32) 1 On substituting the values of P i (n) and Q i (n) from equations (18) and (19) in the above equation and after some algebra, we get ) ρρ llδδ ll μμ ll nn =1 ) WW qq = LL SS ββ ρρ 1 ρρ 1 δδ 1 + ll=2( nn =1 rr nn + 1 αα HH 2 1ρρ 1 γγ 2 1 + ll=2( rr nn αα ll ρρ ll γγ 2 ll (33) where ρρ HH is given in equation (24) 5. SENSITIVITY ANALYSIS The primary objective of this section is to examine the accuracy of approximate results established using the MEP. We present numerical comparisons between exact waiting time and approximate waiting for our model for specific distribution. The default parameters chosen are λ=0.5, r 1 =0.2, r 2 =0.8, (α 1, α 2, α 3 ) = (0.01, 0.05, 0.03) for all distributions. For the purpose of sensitivity analysis we consider our model with the following assumptions: FES and two MOS Mean batch size fixed X=2 and X=3 For M/M/1 : (µ 1, µ 2, µ 3 ) = (2, 10, 10) and (β 1, β 2, β 3 ) = (3, 2, 2) For M/E 2 /1 : (µ 1, µ 2, µ 3 ) = (3, 4, 9) and (β 1, β 2, β 3 ) = (1, 2,1) For M/Γ(1.5)/1 ) : (µ 1, µ 2, µ 3 ) = (3, 8, 10) and (β 1, β 2, β 3 ) = (1, 2, 1) Using the above data the assumptions, an efficient Matlab computer program is built to calculate the exact waiting time W q and the approximate waiting time W q * for different distributions and the results are tabulated in Tables 1-4. We observe from Table 1 that by increasing the arrival rate, the exact waiting time and the approximate waiting time increase for all the distributions. It also increases with the increase in mean batch size. The relative percentage error varies from 7-17% in case of exponential distribution, 2-12% for Erlang-2 distribution and 4-12% for gamma distribution. Table 2 exhibits the variation in W q and W q * by increasing the service rates for different distributions. Both the exact and approximate waiting time decrease with the increase in the service rate for all the distributions and both mean batch size 2 and 3, respectively. It is seen the relative percentage error are 7-17% (exponential), 5-14% (Erlang-2) and 5-11% (gamma). Table 3 displays a comparative analysis of waiting times by varying the failure rate of the server. Here again we see that by increasing the failure rate, both W q and W q * increase for all distributions and both mean batch sizes. In case of exponential distribution, the relative percentage error is 7-11% whereas 5-12% for both erlang-2 and Gamma distributions. Lastly, with the increase in repair rate of the server, the exact and approximate waiting times both decrease for all distributions as seen from Table 4. The relative percentage error varies from 7-10% (for exponential), 9-11% (for Erlang-2) and 4-7% (for gamma). Thus we conclude that by keeping low arrival rate and failure rate and increasing the service rate and repair rate, we can minimize the waiting time. Using the MEP, we also observe that approximate W q * is reasonably near the exact solution. It is observed that for calculating the performance measures for complex queues directly the MEP can play a significant role. 6. CONCLUDING REMARKS In this paper, we have applied MEP to find approximate results for waiting time for M X /G/1 model with FES followed by MOS. The server is subject to breakdown while rendering essential or optional services. We perform a comparative analysis between the approximate results obtained using MEP and established exact results. We have shown that the use of MEP is good enough to obtain performance indices of complex congestion situations for practical purposes. MEP is helpful in analyzing many real life situations which otherwise have been too complex to solve analytically. Models M X /M/1 λ E(X)=2 E(X)=3 W q W q * % Error W q W q * % Error 0.2 0.15 0.13 17.82 0.26 0.23 11.89 0.3 0.27 0.23 14.67 0.48 0.44 9.35 2014, IJMA. All Rights Reserved 130
M X /E 2 /1 M X /γ/1 0.4 0.41 0.36 11.84 0.9 0.83 8.25 0.5 0.61 0.55 10.16 1.93 1.78 7.76 0.6 0.93 0.84 9.11 8.5 7.86 7.59 0.2 0.15 0.13 10.67 0.23 0.22 2.05 0.3 0.24 0.23 5.93 0.44 0.43 2.59 0.4 0.37 0.35 4.7 0.85 0.8 5.21 0.5 0.56 0.53 5.08 2 1.83 8.73 0.6 0.87 0.82 6.36 30.58 26.69 12.72 0.2 0.16 0.14 12.88 0.24 0.23 4.51 0.3 0.25 0.23 7.96 0.46 0.44 3.93 0.4 0.39 0.36 6.16 0.86 0.82 5.17 0.5 0.58 0.55 5.8 1.93 1.79 7.18 0.6 0.89 0.83 6.21 12.34 11.16 9.6 Table 1: Comparison of exact waiting time (W q ) and approximate waiting time (W q *) for different distribution by varying arrival rate (λ) Models µ E(X)=2 E(X)=3 W q W q * % Error W q W q * % Error 2 1.61 1.5 7.27 1.93 1.78 7.76 2.5 0.61 0.55 10.16 0.75 0.68 9.48 M X /M/1 3 0.34 0.29 12.88 0.42 0.37 11.12 3.5 0.22 0.18 15.43 0.27 0.24 12.69 4 0.15 0.13 17.82 0.2 0.17 14.18 2 0.56 0.53 5.08 2 1.83 8.73 2.5 0.32 0.29 6.97 0.74 0.68 7.64 M X /E 2 /1 3 0.21 0.19 9.39 0.42 0.38 7.89 3.5 0.15 0.14 11.91 0.28 0.26 8.76 4 0.12 0.1 14.35 0.21 0.19 9.9 2 0.58 0.55 5.8 1.93 1.79 7.18 2.5 0.38 0.36 6.99 0.94 0.88 6.87 M X /γ/1 3 0.28 0.25 8.38 0.59 0.55 7.01 3.5 0.21 0.19 9.87 0.41 0.38 7.42 4 0.17 0.15 11.38 0.31 0.28 8.02 Table 2: Comparison of exact waiting time (W q ) and approximate waiting time (W q *) for different distribution by varying service rate (µ) Models M X /M/1 M X /E 2 /1 α E(X)=2 E(X)=3 W q W q * % Error W q W q * % Error 0.01 0.61 0.55 10.16 1.93 1.78 7.76 0.02 0.62 0.55 10.41 1.97 1.81 8.04 0.03 0.63 0.56 10.65 2.01 1.84 8.31 0.04 0.63 0.56 10.9 2.05 1.88 8.58 0.05 0.64 0.57 11.14 2.09 1.91 8.86 0.01 0.56 0.53 5.08 2 1.83 8.73 0.02 0.58 0.55 5.95 2.16 1.95 9.71 0.03 0.61 0.57 6.8 2.33 2.09 10.66 0.04 0.64 0.59 7.62 2.52 2.23 11.59 0.05 0.66 0.61 8.42 2.73 2.39 12.51 2014, IJMA. All Rights Reserved 131
M X /γ/1 0.01 0.58 0.55 5.8 1.93 1.79 7.18 0.02 0.62 0.58 7.01 2.18 1.99 8.59 0.03 0.67 0.62 8.17 2.46 2.22 9.94 0.04 0.72 0.65 9.28 2.78 2.47 11.25 0.05 0.77 0.69 10.35 3.15 2.75 12.51 Table 3: Comparison of exact waiting time (W q ) and approximate waiting time (W q *) for different distribution by varying failure rate (α) Models 1/γ E(X)=2 E(X)=3 W q W q * % Error W q W q * % Error M X /M/1 M X /E 2 /1 M X /γ/1 1 0.63 0.57 10.58 2.03 1.87 8.29 2 0.62 0.55 10.28 1.95 1.8 7.9 3 0.61 0.55 10.16 1.93 1.78 7.76 4 0.61 0.55 10.11 1.92 1.77 7.69 5 0.61 0.55 10.07 1.92 1.77 7.65 1 0.23 0.2 11.64 0.46 0.42 10.3 2 0.22 0.2 11.39 0.45 0.41 9.92 3 0.22 0.2 11.29 0.45 0.4 9.79 4 0.22 0.2 11.24 0.45 0.4 9.73 5 0.22 0.2 11.21 0.45 0.4 9.69 1 0.58 0.55 5.8 1.93 1.79 7.18 2 0.55 0.52 5.17 1.8 1.69 6.45 3 0.55 0.52 4.95 1.77 1.66 6.2 4 0.54 0.52 4.85 1.75 1.65 6.08 5 0.54 0.52 4.78 1.74 1.64 6.01 Table 4: Comparison of exact waiting time (W q ) and approximate waitingtime (W q *) for different distribution by varying repair rate (β) REFERENCES 1. Arizono, L., Cui, Y. and Ohta, H. (1991): An analysis of M/M/S queueing systems based on the maximum entropy principle, J. Oper. Res. Soc., Vol. 42, 69-73. 2. Avi-Itzhak, B. and Naor, P. (1963): Some queueing problems with service station subject to breakdown, Oper. Res. Vol. 10, 303-320. 3. Bard, Y. (1980): Estimation of state Probabilities using the maximum entropy principle, IBM J. Res. Develop., Vol. 24, 563-569. 4. Choudhary, M.L. and Templeton, J.G.C. (1983): A First Course in Bulk Queues, John Wiley and Sons, New York. 5. Choudhury, G. and Tadj, L. (2011): The optimal control of an unreliable server queue with two phases of service and Bernoulli vacation schedule, Math. & Comp. Mod., Vol. 54, No. 1 2, 673-688. 6. Choudhury, G. and Madan K.C. (2005): A two- stage batch arrival queueing system with a modified Bernoulli Schedule vacation under N-Policy, Math. Comp. Model., Vol. 42, 71-85. 7. Choudhury, G. and Paul, M. (2006): A two phases queueing system with Bernoulli vacation schedule under multiple vacation policy, Stat. Meth., Vol. 3, 174 185. 8. El-Attendi, M. A. and Kouvatsos, D.D. (1983): A maximum entropy analysis of the M/G/1 and G/M/1 queueing systems at equilibrium, Acta Inform., Vol. 19, 339-355. 9. Guiasu, S. (1986): Maximum entropy condition in queueing theory, J. Oper. Res. Soc., Vol. 37, 293-301. 10. Jain, M. and Jain, A. (2006): Principle of maximum entropy for G/G/1 queue with vacation under N-policy, IJICS, Vol. 9, No. 1, 28-37. 11. Jaynes, E.T. (1957):Information theory and statistical mechanics, Phys. Rev., Vol. 106, No. 4, 620 630. 12. Ke, J. C. (2007):Batch arrival queues under vacation policies with server breakdowns and startup/closedown times, Appl. Math. Model., Vol. 31, No.7, 1282-1292. 13. Ke, J. C. (2008): An M X /G/1 system with startup server and J additional options for service, Appl. Math. Model., Vol. 32, No.4, 236-244. 2014, IJMA. All Rights Reserved 132
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