Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 123 https://internationalpubls.com Impatient Customer Queue with Single Working Vacation A.Saila kumari1 * K. V. S. Nagalakshmi 2 1. .A.Saila kumari,Associate Professor,Department of Mathematics , JNTUA college of engineering,Anantapur-515002 2. Research Scholar and Lecturer in Mathematics, Govt. Degree College, Visakha Women’s, Visakhapatnam Article History: Received: 28-10-2024 Revised: 18-11-2024 Accepted: 19-12-2024 Abstract: This paper analyzes an infinite buffer M/M/1 queue with single working vacation in which customers arrive according to a Poisson process. As soon as the system becomes empty, the server takes working vacation. The service rate during regular busy period, working vacation period and vacation times are assumed to be exponentially distributed. During working vacation customers may renege due to impatience and the impatient timer follows exponential distribution. The steady-state probabilities and performance measures are obtained by using probability generating function. Various numerical results are presented to show the effect of the some parameters on the system performance measures. Keywords: Queue, Single working vacation, Reneging, Probability generating function. 1 Introduction The queueing model with server vacations has been well studied in the past three decades and successfully applied in many areas such as manufacturing and production systems, computer and communication systems, etc. Excellent surveys on the earlier works of vacation models have been reported by Doshi [2], Tian and Zhang [9], Ke et al. [4] and so on. Working vacation (π‘Šπ‘‰) is one kind of vacation policy under which the server provides service at a lower speed during the vacation period rather than stopping service completely. It was introduced by Servi and Finn [7] in an 𝑀/𝑀/1 queueing system. The π‘Šπ‘‰ policies are classified as single working vacation (π‘†π‘Šπ‘‰) and multiple working vacation (π‘€π‘Šπ‘‰). Under the π‘†π‘Šπ‘‰ policy the server enters into vacation, when there are no customers in the queue and serves at a lower rate to the . Meanwhile, he only takes one each time, and must come back to the normal working level no matter whether there are customers at the vacation ending instant. If there are customers at the vacation ends, the server begins to serve one customer at the normal rate immediately; otherwise, he will stay in an idle period. Wu and Takagi [12] generalized Servi and Finn [7] 𝑀/𝑀/1/π‘Šπ‘‰ queue to an 𝑀/𝐺/1/π‘Šπ‘‰ queue. Baba [1] analyzed a 𝐺/𝑀/1 queue with π‘€π‘Šπ‘‰. Tian et al. [10] studied and 𝑀/𝑀/1 queue with π‘†π‘Šπ‘‰ using matrix geometric method. Li and Tian [5] generalized this study to 𝐺𝐼/𝑀/1 queue with π‘†π‘Šπ‘‰ and obtained the queue length distributions at pre-arrival and arbitrary epochs using matrix geometric approach. Impatient is the most prominent characteristic as individuals always feel anxious and impatient during waiting for service in real life. The customers’ impatient acts should be involved in the study of queueing system to model real situations exactly. Intermittent operation of a service can be economically appealing whenever full time service would result in significant server idle time or would preclude the use of the server in some other productive capacity. On the other hand, having the Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 124 https://internationalpubls.com server in operative for periods of time may increase the probability of customer loss due to reneging. An 𝑀/𝑀/1 queue with customers reneging has been discussed in Haight [3]. Laxmi et al.[6] presented the analysis of finite buffer Markovian queue with balking, reneging, and working vacations. Selvaraju and Goswami [8] studied impatient customers in an 𝑀/𝑀/1 queue with single and multiple working vacations. Vijaya Laxmi and Jyotsna [11], analyzed the Impatient customer queue with Bernoulli schedule vacation interruption. This paper is an extension of the earlier of the earlier works Tian et al. [10] by including reneging. Using probability generating functions, explicit expressions of the steady-state probabilities are obtained. Various performance measures such as the average system length, average reneging rate are discussed with these performance measures, we demonstrate the parameter effect on the performance indices of the system. The rest of the paper is organized as follows. Section 2 presents description of the model. The explicit expressions of the steady-state probabilities are obtained in Section 3. A variety of numerical results are presented in section 4 followed by conclusion in Section 5. 2 Model description Let us consider an infinite buffer single server queueing system in which customers arrive according to a Poisson process with rate πœ† and server can serve only one customer at a time with an exponential rate πœ‡. The server begins a working vacation of random length 𝑉 at the instant when the queue becomes empty, and vacation duration 𝑉 follows an exponential distribution with parameter πœ™ . During a working vacation an arriving customer is served at a rate πœ‚. When a vacation ends, if there are customers in the queue, the server changes service rate from πœ‚ to πœ‡ and a regular busy period starts. Otherwise the server enters idle period, and a new regular busy period starts when a customer arrival occurs. An arriving customer who joins the system and finds the server in π‘Šπ‘‰, activates an impatient timer 𝑇, which is exponential with rate 𝛼. If the server is available during π‘Šπ‘‰ before the time 𝑇 expires, the customer is served with πœ‚. If π‘Šπ‘‰ finishes before the impatient timer expires, the server switches to normal working period, and the customer is served with rate πœ‡. Since the arrival and departure of an impatient customer without service are independent the average reneging rate of a customer is given by 𝑛𝛼, where 𝑛 is the number of customers. 3 Steady-state probabilities At time 𝑑, let 𝐿(𝑑) denote the number of customers in the system and 𝐽(𝑑) be the status of the server, which is defined as 𝐽(𝑑) = 0, when the server is in π‘Šπ‘‰ and 𝐽(𝑑) = 1, when server is in normal working period. The {(𝐽(𝑑), 𝐿(𝑑)), 𝑑 β‰₯ 0} is a Markov process with the state space Ξ© = {(𝑗, 𝑛): 𝑛 β‰₯ 0, 𝑗 = 0,1}. Let πœ‹π‘—,𝑛 = lim π‘‘β†’βˆž πœ‹{𝐽(𝑑) = 𝑗, 𝐿(𝑑) = 𝑛}, 𝑛 β‰₯ 0, 𝑗 = 0,1 , denote the steady-state probabilities of the process {(𝐽(𝑑), 𝐿(𝑑)), 𝑑 β‰₯ 0}. Then, the set of balance equations are given as follows: (+)_0,0 = (+)_0,1+_1,1, (+++)_0,1 = _0,0+(+2)_0,2, (+++n)_0,n = _0,n-1+(+(n+1))_0,n+1; n 2, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 125 https://internationalpubls.com _1,0 = _0,0, (+)_1,1 = _1,0+_1,2+_0,1, (+)_1,n = _1,n-1+_1,n+1+_0,n; n2. The steady state probabilities are obtained by solving the above system of equations (3)-(3) using probability generating function. Define 𝐺0(𝑧) and 𝐺1(𝑧) as 𝐺0(𝑧) = βˆ‘βˆž 𝑛=0 π‘§π‘›πœ‹0,𝑛 , 𝐺1(𝑍) = βˆ‘βˆž 𝑛=0 π‘§π‘›πœ‹1,𝑛 , with 𝐺0(1) + 𝐺1(1) = 1. and 𝐺0 β€² (𝑧) = (𝑑/𝑑𝑧)𝐺0(𝑧) = βˆ‘βˆž 𝑛=1 π‘›π‘§π‘›βˆ’1πœ‹0,𝑛 . Multiplying equations (3)-(3) by 1, 𝑧 , 𝑧𝑛 ; respectively, summing over n and rearranging the terms. We get 𝛼𝑧(1 βˆ’ 𝑧)𝐺0 β€² (𝑧) + (πœ†π‘§2 βˆ’ (πœ† + πœ™ + πœ‚)𝑧 + πœ‚)𝐺0(𝑧) + πœ‡πœ‹1,1𝑧 βˆ’ (1 βˆ’ 𝑧)πœ‚πœ‹0,0 = 0. (1) In similar manner, from equations (3)-(3), we have (πœ†π‘§ βˆ’ πœ‡)(1 βˆ’ 𝑧)𝐺1(𝑧) = πœ™π‘§πΊ0(𝑧) βˆ’ πœ‡(1 βˆ’ 𝑧)πœ‹1,0 βˆ’ πœ‡πœ‹1,1𝑧. (2) From equation (1) can be written as 𝐺0 β€² (𝑧) βˆ’ ( πœ† 𝛼 + (πœ™+πœ‚) 𝛼(1βˆ’π‘§) βˆ’ πœ‚ 𝛼𝑧(1βˆ’π‘§) ) 𝐺0(𝑧) = πœ‚ 𝛼𝑧 πœ‹0,0 βˆ’ πœ‡ 𝛼(1βˆ’π‘§) πœ‹1,1 for 𝑧 β‰  0 and 𝑧 β‰  1. Multiplying with π‘’βˆ’(πœ†/𝛼)𝑧(1 βˆ’ 𝑧)πœ™/π›Όπ‘§πœ‚/𝛼 on both sides in the above equation and integrating from 0 to 𝑧 yields, G_0(z)=-_1,1k_1(z)+_0,0k_2(z) e^-(/)z(1-z)^/ z^/, where k_1(z)=_0^ze^-(/)x(1-x)^(/)-1x^/dx, k_2(z)=_0^ze^-(/)x(1-x)^/x^(/)-1dx. Let 𝐺0(1) and 𝐺1(1) denotes the sum of probabilities when the server is in working vacation and normal busy period and define 𝐺0(1) = βˆ‘βˆž 𝑛=0 πœ‹0,𝑛 and 𝐺1(1) = βˆ‘βˆž 𝑛=0 πœ‹1,𝑛, respectively. Taking 𝑧 = 1 in equation (3), we get πœ‡πœ‹1,1π‘˜1(1) = πœ‚πœ‹0,0π‘˜2(1). (3) Therefore equation (3) becomes as 𝐺0(𝑧) = 𝑒(πœ†/𝛼)𝑧 (1βˆ’π‘§)πœ™/π›Όπ‘§πœ‚/𝛼 [π‘˜2(𝑧) βˆ’ π‘˜2(1) π‘˜1(1) π‘˜1(𝑧)] πœ‚ 𝛼 πœ‹0,0. (4) The average number of customers in the system during a working vacation is dinotes 𝐸[𝐿0] and define 𝐸[𝐿0] = 𝐺0 β€² (1) = βˆ‘βˆž 𝑛=1 π‘›πœ‹0,𝑛. From the equation(1), 𝐺0 β€² (𝑧) = (1βˆ’π‘§)πœ‚πœ‹0,0βˆ’(πœ†π‘§2βˆ’(πœ†+πœ™+πœ‚)𝑧+πœ‚)𝐺0(𝑧)βˆ’πœ‡πœ‹1,1𝑧 𝛼𝑧(1βˆ’π‘§) . Take 𝑧 = 1 and using L’Hospital rule, 𝐺0 β€² (1) can be obtained as 𝐺0 β€² (1) = (πœ†βˆ’πœ‚)𝐺0(1)+πœ‚πœ‹0,0 (𝛼+πœ™) . (5) Now we obtained 𝐺0(1) and πœ‹0,0. Taking 𝑧 = 1 in equation (2), we have 𝐺0(1) = πœ‡πœ‹1,1 πœ™ . (6) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 126 https://internationalpubls.com From equation (2), 𝐺1(𝑧) = πœ™π‘§πΊ0(𝑧)βˆ’πœ‡(1βˆ’π‘§)πœ‹1,0βˆ’πœ‡πœ‹1,1𝑧 (πœ†π‘§βˆ’πœ‡)(1βˆ’π‘§) , and using L’Hospital rule by taking 𝑧 = 1, then 𝐺1(1) can be obtained as 𝐺1(1) = πœ™πΊ0 β€² (1)+πœ‡πœ‹1,0 (πœ‡βˆ’πœ†) . (7) From the equations (3) and (6), 𝐺0(1) can be written as 𝐺0(1) = π›½πœ‹0,0, where 𝛽 = πœ‚π‘˜2(1) πœ™π‘˜1(1) . Using equation (3) and the normalizing condition i.e. 𝐺0(1) + 𝐺1(1) = 1, we get πœ‹0,0 = πœ†=(𝛼+πœ™)(πœ‡βˆ’πœ†) (𝛼+πœ™)(π›½πœ†(πœ‡βˆ’πœ†)+πœ‡πœ™)+πœ†πœ™(𝛽(πœ†βˆ’πœ‚)+πœ‚) . (8) 4 Performance measures In this section, we present some important performance measures of the model. Let 𝐸[𝐿1] denote the average number of customers in the system when the server is in normal working period. Using equation (3), and Differentiating equation (2) with respect to 𝑧 and using L’Hospital rule twice, the expression for 𝐸[𝐿1] is obtained as 𝐸[𝐿1] = 𝐺1 β€² (1) = πœ™πΊβ€²β€²0(1) 2(πœ‡βˆ’πœ†) + πœ‡πœ™ (πœ‡βˆ’πœ†)2 𝐸[𝐿0] + πœ‡πœ™ (πœ‡βˆ’πœ†)2 πœ‹0,0. (9) Differentiating equation (1) with respect to 𝑧 and using L’Hospital rule,taking 𝑧 = 1, we have 𝐺′′0(1) = 2((πœ†βˆ’πœ™βˆ’πœ‚βˆ’π›Ό)𝐸[𝐿0]+πœ†πΊ0(1)) 2𝛼+πœ™ , substituting 𝐺′′0(1) and 𝐸[𝐿0] in equation (4) yields 𝐸[𝐿1]. Therefore, the average system length 𝐸[𝐿] is given by 𝐸[𝐿] = 𝐸[𝐿0] + 𝐸[𝐿1]. As the instantaneous reneging rate during π‘Šπ‘‰ is 𝑛𝛼, the average reneging rate (𝑅. 𝑅) is 𝑅. 𝑅 = βˆ‘βˆž 𝑛=1 π‘›π›Όπœ‹0,𝑛 = 𝛼𝐸[𝐿0] 5 Numerical results In this section, we present the effect of some parameters on the system performance measures through some numerical results. The various parameters of the model are chosen as πœ† = 1.7; πœ‡ = 2.5; πœ‚ = 1.9; πœ™ = 1.5; 𝛼 = 1.1; unless they are considered as variables or their values are mentioned in the respective figures. Figure 1: Effect πœ† on 𝐸[𝐿] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 127 https://internationalpubls.com Figure 2 presents the effect of arrival rate (πœ†) on the average system length (𝐸[𝐿]) for different vacation rates (πœ™). 𝐸[𝐿] increase with the increase of πœ† for fixed πœ™ and it decreases with the increase of πœ™ for fixed πœ† as it should be. Figure 2: Effect 𝑅. 𝑅. for different πœ™ Figure 2 shows the effect of 𝛼 on the reneging rate (𝑅. 𝑅. ) for different vacation rates (πœ™). As expected in real life, 𝑅. 𝑅. increases with the increase of 𝛼. In this paper we considered that, the arrived customer may renege due to impatience whenever the server is in π‘Šπ‘‰. However, the server spends more time in normal working period with the increase of πœ™ and hence 𝑅. 𝑅. decrease with the increase of πœ™ for fixed πœ†. We can clearly observe this case from Figure 2. Figure 3: Effect πœ‚ on 𝐸[𝐿] for different πœ™ The effect of service rates in working vacation period (πœ‚) on the average system length 𝐸[𝐿] for different vacation rates πœ™ is given in Figure 3. We know that, 𝐸[𝐿] decreases with πœ‚ for fixed πœ™ and it increases with the increase of πœ™ for fixed πœ‚ as it clear from real life. Figure 4: Effect πœ‡ on 𝐸[𝐿] for different πœ† Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 128 https://internationalpubls.com Figure 4 gives the effect of service rates in normal working period (πœ‡) on the average system length 𝐸[𝐿] for different arrival rates πœ†. As explained in Figure 2, 𝐸[𝐿] increases with πœ† for fixed πœ‡. Also it decreases with the increase of πœ‡ for fixed πœ†. Figure 5: Effect πœ‚ on 𝐸[𝐿] for different πœ† Figure 5 depicts the effect of service rates πœ‚ on the average system length 𝐸[𝐿] for different arrival rates πœ† . Here also 𝐸[𝐿] increases with the increase of πœ† for fixed πœ‚ and decreases with increase of πœ† for fixed πœ‚ as in Figure 4. But, the effect of service rate on 𝐸[𝐿] is more in Figure 4 when compare with Figure 5. This is due to the service rate in working vacation period is lower than the service rate in normal working period. 6 Conclusion In this paper, we have studied an 𝑀/𝑀/1 queueing system with single working vacation and customers’ impatience. We have obtained the closed form expressions of the steady state probabilities. Some important performance measures of the model such as average system length and average reneging rate. The effects of some parameters on the performance measures of the system have been investigated numerically and presented through graphically. The techniques adopted in this paper can be applied to analyze models like impatient customer 𝑀/𝑀/1 queue with variant working vacation. References [1] Baba, Y. (2005) Analysis of a 𝐺𝐼/𝑀/1 queue with multiple working vacations. Operation Research Lett 33, pp. 201–9. [2] Doshi, B.T., (1986) Queueing systems with vacations-a survey. Queueing Systems 1, pp. 29–66. [3] Haight, F. A., (1959) Queing with Reneging. Metrika 2, pp.186–197. [4] Ke, J.C., Wu, C. H., and Zhang, Z.G., (2010) Recent developments in vacation queueing models: a short survey.International Journal of Operation Research 7, pp. 3–8. [5] Li, J. and Tian, N. (2011) Performance Analysis of a 𝐺𝐼/𝑀/1 queue with single working vacation. Applied Maths Computation 217, pp. 4960–71. [6] Vijaya Laxmi, P., Goswani, V. and Jyothsna, K. (2013) Analysis of finite buffer Markovian queue with balking, reneging and working vacations. International Journal of Strategic Decision Sciences 4, pp. 1–24. 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