Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 312 https://internationalpubls.com Optimality of Finite Capacity Markovian Queues with Discouraged Arrivals and Single hiatus with Waiting Server J. Vimal Andrew1, V. Vijayalakshmi*2 1Department of Mathematics, Karpagam Academy of Higher Education, Coimbatore - 641 021, Tamil Nadu, India. 2Department of Science and Humanities (Mathematics), Faculty of Engineering, Karpagam Academy of Higher Education, Coimbatore-641 021, Tamil Nadu, India. 1vimalandmat@gmail.com ∗2viji.sanmugam@gmail.com Article History: Received: 15-05-2024 Revised: 25-06-2024 Accepted: 10-07-2024 Abstract: We consider finite-capacity Markovian queues with a single hiatus scheme and waiting server. Customers are arriving at a Poisson arrival λ and exponential service distribution, with a mean service rate µ. In which customers join the queue according to the number of customers in the system while the hiatus is in the service-providing process. For the assumed queuing model, steady-state probabilities were derived, and some important performance measures, such as the mean number of customers in the system and mean response time in the system and queue are analysed. The expected expense function is developed and formulated as an optimization problem in order to find the minimum expense. Numerical illustrations are given to show the effect of parameters on the performance measures. Keywords: PGF, Vacations, Performance measures, PSO. Mathematics Subject Classification: 90B22 and 60K25. 1. Introduction In this paper, we consider a single server queueing system where the service time of each customer depends on the number of customers served prior to him in the current busy period and with server vacation. Several researchers have studied queueing systems in which the service time of a customer depends on the number of customers served in the current busy period. Doshi [5], Takagi [16] and Tian and Zhang [17] are excellent survey works on the subject. Teghem [10] has made a comprehensive survey of queueing system with vacation. Mishra et al. [19] discussed and investigated transient behavior of a M/M/1 waiting line undergoing multiple differentiated vacations in conjunction with impatient customers- manifested in the form of balking and probabilistically modified reneging. Vijayalakshmi et al. [20] discussed about arriving customers to receive only one service and may want to choose some optional service from the services available in the system. Baburaj and Sahana [21] studied the discrete time single arrival and single-batch service queue under policy ’C’ is de- scribed in this study together with reneging and vacation interruption. Yumei Hou et al. [22] discussed optimization of beds allocation based on queuing model and by Particle Swarm Optimization Algorithm. According to Ammar et al. [1], customers balk using a set probability and renege based on a negative exponential distribution. Bouchentouf et al. [3] examined a single-server M/M/1/N feedback queuing system that includes vacation, balking, reneging and customer retention. Abdul Rasheed et al. [2] investigated the discouraged arrival of markovian queueing systems, which regulate service speed according on customer count. Courtois and George [4] found that a customer’s desire for service is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 313 https://internationalpubls.com influenced by the duration of their wait time. Haddadi [6] proposed a method for studying busy period processes in queue models. Satish Kumar et al. [7] developed a single-server Markovian queueing system with limited capacity, encouraged or discouraged arrivals and a modified customer reneging policy. Reynolds [8] researched multi-server queuing models with diminishing arrival rates as queue length increases. Kumar [9] studied a reneged customer may be persuaded to stay in the wait for additional service with a probability of q or they may depart the queue without obtaining service with a probability of p = 1 -q. Parthasarathy et al. [11,12] conducted a transient analysis of a line with potential customers discouraged by its length and a fluid queue driven by discouraged arrivals. They obtained explicit formulas for the stationary distribution function. Sanga et al. [13] proposed a queueing model with a single server, finite capacity, discouraged customers, and distributed retry times. Sharma et al. [14] computed closed-form probability for transient states in a finite waiting space. Rao [15] mentioned, the M/G/1 queuing procedure involves units that balk and renege. Unit servicing may experience breakdowns due to disruptions, which must be addressed promptly. Van Doorn [18] investigated exact formulas for the birth-death process transition probability. The model treated in this paper is immediately applicable to many fields such as computer systems, telecommunication systems and production systems. The outlook of this paper is as follows. We describe the model and introduce notation in section 2. In section 3, we obtain the steady state probabilities and the moments of the queue length distribution. The performance measures are obtained in section 4. Optimization process is carried out in section 5 by Total cost method, Direct search method and particular swarm optimization (PSO)method. Numerical analysis is carried out in section 6. 2. Model Formulation Consider a finite buffer Markovian queue with a single hiatus policy with the following assumptions. • Beneficiary entries occur in a Poisson stream at a rate λ • The service times follow an exponential distribution with µ. • The beneficiaries are receiving service on a First Come First Serve (FCFS). • A single hiatus(α) scheme is followed. Once the system reaches the zero-beneficiary level, the service provider departs from the service station for a hiatus. • If there are beneficiaries in the waiting line at the end of a hiatus, the server begins to provide service. Otherwise, the service provider remains waiting in the service station for the new beneficiaries. A hiatus period is exponentially distributed with γ. • While the service provider is on hiatus, arriving beneficiaries are discouraged harmonically with respect to the number of beneficiaries waiting in system, that is, 𝜆𝑛 = 𝜆 𝑛+1 , 𝑛 ≥ 0 Figure 1: State transition diagram Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 314 https://internationalpubls.com • The state transition diagram of the queueing model undertaken is displayed in figure 1. • Let N(t) be the quantity of customers in the system at time t. At that point the bivariate process {(C(t), N(t)), t ≥ 0} is a persistent time Markov chain. Let Pi,n(t) = Prob {C(t) = i, N(t) = n, i = 0, 1and n ≥ 0} 𝐶(𝑡) = { 0, 𝑖𝑓 𝑎 𝑣𝑎𝑐𝑎𝑡𝑖𝑜𝑛 𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑡𝑎𝑘𝑒𝑠 𝑝𝑙𝑎𝑐𝑒 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡 1, 𝑖𝑓 𝑡ℎ𝑒 𝑠𝑒𝑟𝑣𝑒𝑟 𝑖𝑠 𝑖𝑑𝑙𝑒 (𝑜𝑟) 𝑎 𝑟𝑒𝑔𝑢𝑙𝑎𝑟 𝑠𝑒𝑟𝑣𝑖𝑐𝑒 𝑡𝑎𝑘𝑒𝑠 𝑝𝑙𝑎𝑐𝑒 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡 } 3. Steady State Analysis By Markov theory, the steady state balance flow equations of assumed model are as follows. (𝜆 + 𝛾)𝑝0,0 = α𝑝1,0 (1) ( 𝜆 𝑛+1 + 𝛾) 𝑝0,𝑛 = 𝜆 𝑛 𝑝1,𝑛−1, 1 ≤ 𝑛 ≤ 𝑁 − 1 (2) 𝛾𝑝0,𝑁 = 𝜆 𝑁 𝑝0,𝑁−1 (3) (𝜆 + 𝛼)𝑝1,0 = γ𝑝0,0 + µ𝑝1.1 (4) (𝜆 + µ)𝑝1,𝑛 = λ𝑝1,𝑛−1 + µ𝑝1.𝑛+1+ϒ𝑝0,𝑛 , 1 ≤ 𝑛 ≤ 𝑁 − 1 (5) (𝜆 + µ)𝑝1,𝑛= 𝜆𝑝1,𝑁−1 + 𝛾𝑝0,𝑁 (6) Let 𝑃𝑖(𝑧) = ∑ 𝑝𝑖,𝑛 𝑁 𝑖=0 𝑧𝑛, 𝑖 = 0,1 (7) be the partial probability generating functions. From (3), 𝑝0,𝑁 = 𝜆 𝑁𝛾 𝑝0,𝑁−1 The recursive application of (2) becomes 𝑝0,𝑁 = 𝜆𝑁 𝛾 ∏ (𝜆+𝑖𝛾)𝑁 𝑖=2 (8) Multiply (4) - (6) by appropriate 𝑧𝑛 and adding, we have 𝑃1(𝑍) = 𝜆𝑧𝑁(1−𝑧)𝑝1,𝑁−𝛼𝑝1,0+µ𝑝1,0 +𝛾𝑃0(𝑍) − µ𝑝1,0 𝑧 𝜆(1−𝑧)− µ 𝑧 +µ (9) Here the denominator of 𝑃1(𝑧) has tworoots 1, µ 𝜆 From this 𝑃0(𝑍) = 𝛼 𝛾 𝑝1,0 (10) and 𝑧 = µ 𝜆 is also the root of the numerator of P1(z) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 315 https://internationalpubls.com 0 = 𝜆( µ 𝜆 )𝑁(1 − µ 𝜆 )𝑝1,𝑁 − 𝛼𝑝1,0 + µ𝑝1,0 + 𝛾𝑝0( µ 𝜆 ) − 𝜆𝑝1,0 (11) From (1), (2) and (3) 𝑃0(𝑍) = 𝑝0,0 [1 + ∑ (𝑛+1)(𝜆𝑧)𝑛 ∏ (𝜆+(𝑖+1)𝛾)𝑛 𝑖=1 𝑁−1 𝑛=1 + (𝜆𝑧)𝑁 ∏ (𝜆+𝑖𝛾)𝛾𝑁 𝑖=2 ] (12) Substitute 𝑧 = µ 𝜆 in the above equation, we have 𝑃0( µ 𝜆 ) = 𝑝0,0 Ө (13) where Ө = 1 + ∑ (𝑛 + 1) ∏ 𝜆𝑖 𝜆+𝑖𝛾 𝑛+1 𝑖=2 𝑁−1 𝑛=1 ( µ 𝜆 )𝑛 + 1 𝛾 ∏ 𝜆𝑁 𝜆+𝑖𝛾 𝑁 𝑖=2 ( µ 𝜆 )𝑁 From (11) and (13) we have 𝑝1,𝑁 = (𝛼−µ+𝜆) 𝜆+µ 𝛼 +𝛾Ө (𝜆−µ) µ 𝜆 𝑝0,0 (14) From the local balance equation, ρ𝑝1,0+ ρ 1 𝑝0,0 = 𝑝1,1 ρ𝑝1,1+ ρ 1 𝑝0,1 = 𝑝1,2 . . . ρ𝑝1,𝑁−1+ ρ 𝑁 𝑝0,𝑁−1 = 𝑝1,𝑁 Generally ρ𝑝1,𝑛+ ρ 𝑛 𝑝0,𝑁−1 = 𝑝1,𝑛+1 , 0≤n≤N-1 (15) After some mathematical manipulations equation (15), we obtain 𝜌[𝑃1(z)-𝑝1,𝑁𝑧𝑁]+ρ∑ 1 𝑛 𝑁 𝑛=1 𝑝0,𝑛−1= 1 𝑧 [𝑃1(z)-𝑝1,0] put 𝑧 = 1 𝑃1(1) = 𝜌 𝜌−1 𝑝1,𝑁 − 𝜌 𝜌−1 ∑ 1 𝑛 𝑁 𝑛=1 𝑝0,𝑛−1 − 1 𝜌−1 𝑝1,0 (16) From the law of total probability, 𝑝0,0 = 1 𝜆+𝛾 𝛾 + 𝜌 𝜌−1 [𝜌𝑁( 𝜆+𝛾+𝛼 𝛼 )+ 𝜆𝜌 𝜆+2𝛾 (𝜌𝑁−2+∑ 𝜆𝑖−2𝜌𝑁−𝑖 ∏ (𝜆+𝑗𝛾)𝑖 𝑗=3 )]− 𝜆+𝛾 (𝜌−1)𝛼 − 𝜌 𝜌−1 ∑ 𝑁𝜆𝑁−1 𝑛 ∏ (𝜆+𝑖𝛾)𝑁 𝑖=2 𝑁 𝑛=1 𝑁 𝑖=3 (17) Substitute the value of 𝑝0,0 in (1), Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 316 https://internationalpubls.com 𝑝1,0=( 𝜆+𝛾 𝛼 ) 𝑝0,0 𝑝1,0=( 𝜆+𝛾 𝛼 ) 1 𝜆+𝛾 𝛾 + 𝜌 𝜌−1 [𝜌𝑁( 𝜆+𝛾+𝛼 𝛼 )+ 𝜆𝜌 𝜆+2𝛾 (𝜌𝑁−2+∑ 𝜆𝑖−2𝜌𝑁−𝑖 ∏ (𝜆+𝑗𝛾)𝑖 𝑗=3 )]− 𝜆+𝛾 (𝜌−1)𝛼 − 𝜌 𝜌−1 ∑ 𝑁𝜆𝑁−1 𝑛 ∏ (𝜆+𝑖𝛾)𝑁 𝑖=2 𝑁 𝑛=1 𝑁 𝑖=3 (18) Figure 2: P1,0 against λ Figure 3: P1,0 against λ Figure 4: P1,0 against λ As a regular service period ends, if there are customers in the system, the system will be resumed to the regular service period. Otherwise, the server will enter into a vacation, during which service is not rendered to any of new arrivals in the period completely. Ultimately as the arrival rate λ increases, the steady state probability which is in equation (18) decreased which is shown in Figure 2, for the fixed values of µ = 0.51 and γ = 0.7, by varying the values of λ from 0.1 to 0.5 with different values of α = 0.5, 0.6, 0.7. As aforesaid, in Figure 3, the nature of decreasing continues in the steady state probability for the fixed values of µ = 0.51 and α = 0.8, by varying the values of λ from 0.1 to 0.5 with different values of γ = 0.6, 0.7, 0.8. This nature continues in Figure 4, for the fixed values of γ = 0.8 and α = 0.9 by Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 317 https://internationalpubls.com i varying the values of λ from 0.1 to 0.5 with different values of µ = 0.51, 0.6, 0.7. 4. Performance Measure Expected number of customers in the system 𝐸(𝐿) = 𝑝0,0 {∑ 𝑖(𝑖 + 1)𝜆𝑖 ∏ (𝜆 + (𝑗 + 1)𝛾)𝑖 𝑗=1 + 𝑁𝜆𝑁 𝛾 ∏ (𝜆 + 𝑖𝛾)𝑁 𝑖=2 𝑁 𝑖=1 + 𝜌 ( 𝜆 + 𝛾 + 𝛼 𝛼 ) + ∑ [𝑗𝜌𝑗 ( 𝜆 + 𝛾 + 𝛼 𝛼 ) + 𝜆 𝜆 + 2𝛾 (𝜌𝑗−2 + ∑ 𝜆𝑘−2𝜌𝑗−𝑘 ∏ (𝜆 + 𝑙𝛾)𝑘 𝑙=1 𝑗 𝑘=3 )] 𝑁 𝑗=2 } where p0,0 is already in (17) and the expected waiting time is 𝐸(𝑊) = 𝐸(𝐿) 𝜆 The graph for the probability P1,0 against λ, E(L) and E(W) for distinct parame ters are as follows. Figure 5: E(L) against λ Figure 6: E(L) against λ Figure 7: E(L) against λ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 318 https://internationalpubls.com As the arrival rate λ increases then the number of customers in the line is also increased during the vacation period. In Figure 5, for the fixed values of µ = 0.7 and γ = 0.6, by varying the values of λ from 0.1 to 0.5 with different values of α = 0.5, 0.6, 0.7. The value of E(L), Figure 6 increases for the fixed values of µ = 0.7 and α = 0.6, by varying the values of λ from 0.1 to 0.5 with different values of γ = 0.5, 0.6, 0.7. The expected number of customers in the line is also increased during the vacation period. In Figure 7, for the fixed values of α = 0.55 and γ = 0.65, by varying the values of λ from 0.1 to 0.5 with different values of µ = 0.7, 0.8, 0.9. Figure 8: E(W) against λ Figure 9: E(W) against λ Figure 10: E(W) against λ As the arrival rate λ increases then the number of waiting customers in the waiting line is also increased during the vacation period. In Figure 8, for the fixed values of µ = 0.6 and γ = 0.7, by varying the values of λ from 0.1 to 0.5 with different values of α = 0.5, 0.6, 0.7. The value of E(W) in Figure 9 increases, for the fixed values of µ = 0.6 and α = 0.7, by varying the values of λ from Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 319 https://internationalpubls.com 0.1 to 0.5 with different values of γ = 0.55, 0.65, 0.8. The expected number of waiting customers in the waiting queue is also in- creased during the vacation period. In Figure 10, for the fixed values of α = 0.6 and γ = 0.5, by varying the values of λ from 0.1 to 0.5 with different values of µ = 0.7, 0.8, 0.9. 5. Reliability Measures Because of the importance of dependability measures under varied queueing conditions, a few investigations are dedicated to queueing hypothesis writing. Accessibility is defined as the likelihood that the system will function properly when it is mentioned for use. However, disappointed recurrence of the server is a possibility when the system is on vacation period. In this manner, we obtain the two key dependability measures • The availability of the server 𝑃1(1) = 𝜌𝑝0,0 𝜌 − 1 [𝜌𝑁( 𝜆 + 𝛾 + 𝛼 𝛼 ) + 𝜆𝜌 𝜆 + 2𝛾 (𝜌𝑁−2 + ∑ 𝜆𝑖−2𝜌𝑁−𝑖 ∏ (𝜆 + 𝑗𝛾)𝑖 𝑗=3 )] − 𝜆 + 𝛾 𝜌𝛼 − ∑ 𝑁𝜆𝑁−1 𝑛 ∏ (𝜆 + 𝑖𝛾)𝑁 𝑖=2 𝑁 𝑛=1 𝑁 𝑖=3 ] • The failure frequency of the server 𝑃0(1) = ( 𝜆 + 𝛾 𝛾 ) 𝑝0,0 6. Optimization Analysis 6.1 Total Cost method We develop the cost model to enable the minimum cost over the system. The various cost parameters are defined as CO1 ≡ holding cost for every customer present in the system CO2 ≡ waiting cost for every customer waits in the system CO3 ≡ cost for the server in the busy period CO4 ≡ cost for the server in the vacation period CO5 ≡ cost for service Using the definition of these cost elements listed above, the expected cost function per unit time is given by TC ≡ CO1E(L) + CO2E(W) + CO3P0(Z) + CO4P1(Z) + CO5µ The cost minimization problem can be formulated as TC (µ, γ) = Minimize (µ, γ) subject to µ > γ and ρ < 1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 320 https://internationalpubls.com We consider the following cost parameters as CO1 = 20, CO2 = 45, CO3 =60, CO4 = 80, CO5=90 Figure 11: Total Cost against µ Table 1: Cost function versus µ µ 1 2 3 4 5 6 7 8 9 10 λ = 0.6 650.3 557.1 612.1 687.6 769.8 854.9 941.5 1029.1 1117.2 1205.7 λ = 0.8 835.1 672.5 719.1 792.5 873.9 958.6 1045.1 1132.5 1220.6 1309.1 λ = 0.9 912.2 760.4 802.1 874.5 955.3 1039.9 1126.2 1213.6 1301.6 1390.1 In Figure 11, for the different arrival rate λ = 0.6, 0.8, 0.9 and for the fixed values of γ = 0.1 and α = 0.2, we got the values while varying µ from 1 to 10 as follows the table. The minimum values corresponding to the given values of 𝜆 are 557.1, 672.5 and 760.4 respectively. It is seen that initially the total cost diminishes and begins expanding with the ward of µ for fixed estimations of γ and α. The raised nature of the cost work concerning show the pattern for the ideal expense by expanding the typical assistance sace of the clients. This shows the convex nature. Indeed, it is not possible to derive the analytic solutions for the optimal service rates at the minimum expected cost. Thus, we progress the approximations to achieve the optimal service rates by direct search method 6.2 Direct Search Method We assumed the following cost parameters as CO6 = 30, CO7 = 45, CO8 = 90 The cost element is defined as CO6 ≡ cost for expected number of customers in busy period CO7 ≡ cost for expected number of customers in vacation period CO8 ≡ cost for the server in the busy period Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 321 https://internationalpubls.com The cost function TC is defined as 𝑇𝐶 ≡ 𝐶𝑂6𝑃0 1(𝑍) + 𝐶𝑂7𝑃1 1(𝑍) + 𝐶𝑂8µ Figure 12: Total Cost against µ Table 2: Cost function versus µ µ 1 2 3 4 5 6 7 8 9 10 λ = 0.3 833.1 497.4 479.5 520.2 582.1 653.7 730.9 811.5 894.2 978.4 λ = 0.4 1344.6 668.4 584.6 598.6 645.3 707.3 788.6 854.4 933.6 1015.1 λ = 0.5 2088.4 885.9 713.8 693.1 722.1 773.3 836.6 907.1 982.1 1060.3 In Figure 12, for the different hiatus rate λ = 0.3, 0.4, 0.5 and for the fixed values of α = 0.1 and γ = 0.7 we got the values while varying µ from 1 to 10 as follows the table. The minimum values corresponding to the given values for λ are 479.5, 584.6 and 693.1 respectively. It is evident that total cost function decreases first and then increases. Consequently, the convex nature arises in the total cost function. This confirms the possibility of obtaining the optimum service rates. Figure 13: Total Cost against µ Table 3: Cost function versus µ µ 1 2 3 4 5 6 7 8 9 10 γ = 0.3 1429.4 677.2 596.1 613.1 663.6 728.3 800.8 878.1 958.2 1040.5 γ = 0.5 1683.5 752.7 635.5 637.9 679.6 739.1 807.9 882.4 960.6 1041.3 γ = 0.7 1951.5 893.8 686.1 673.1 706.3 760.3 825.4 897.2 973.3 1052.4 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 322 https://internationalpubls.com In Figure 13, for the values of γ = 0.3, 0.5, 0.7 and for the fixed values of λ = 0.5 and α = 0.1 we got the values while varying µ from 1 to 10 as follows the table. The minimum values corresponding to the given values for γ are 596.1, 635.5 and 673.1 respectively. The figure shows the convexity in the total cost function. The convex nature of the cost function with respect to µ shows the trend for the optimum cost by increasing the normal service domain of the customers. Figure 14: Total Cost against µ Table 4: Cost function versus µ µ 1 2 3 4 5 6 7 8 9 10 α = 0.15 912.5 509.8 487.9 528.8 591.2 663.4 741.1 822.1 905.1 989.5 α = 0.21 717.1 437.4 443.5 496.6 565.9 642.6 723.3 806.5 891.3 977.2 α = 0.29 581.9 387.2 412.6 474.1 548.2 627.9 710.8 795.5 881.5 968.3 In Figure 14, for the different hiatus rate α = 0.15, 0.21, 0.29 and for the fixed values of λ = 0.4 and γ = 0.7 we got the values while varying µ from 1 to 10 as follows the table. The minimum values corresponding to the given values of α are 487.9, 437.4 and 387.2 respectively. The figure shows the convexity in the total cost function. This confirms the possibility of obtaining the optimum service rate. 6.3 Particular Swarm Optimization Particle swarm optimization is one of the most popular nature-inspired meta- heuristic optimization algorithms developed by James Kennedy and Russell Eberhart in 1995.Particle swarm optimization (PSO) is inspired by social and cooperative behavior displayed by various species to fill their needs in the search space. Recently, PSO has emerged as a promising algorithm in solving various optimization problems in the field of science and engineering Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 323 https://internationalpubls.com Figure 15: PSO Total Cost against Iteration Table 5: PSO-Effect of µ∗ on TC∗, E(L), E(W) for different value of λ The snapshot of the optimized solution of the problem is shown in the above table. It is observed that the best value of the objective function obtained after 10 independent runs is 48.1851. This value is obtained at x (0.4) = 1.4473. Out of the 10 runs, 10th run gives this best results. The simulation total time taken is 329.2360 seconds. It is to be noted that the simulation time depends on computer configuration. Further, Figure 15 shows the convergence characteristic of PSO. 7. Conclusion As the service rate µ grows, the considered probabilities drop. This is apparent because growing signifies a higher chance of customers abandoning the system during the service receiving stage, resulting in a shorter system length. The purpose of this experiment is to investigate the effect of increasing the arrival rate λ on the average response time µ of a client. Figure 4 shows that as grows, the average waiting time for clients in the system increases, as expected. The purpose of this experiment is to evaluate the behavior of the mean system length versus the vacation rate for the three possibilities of the relationship between λ and µ. Figure 5 shows that the following are true: • With the exception of the case λ < µ, the mean system size increases slowly with increasing. • As the for the example increases, the mean system size falls steadily. This is due to the fact that as the vacation rate γ increases, the server returns to the system sooner. As a result, the predicted number of consumers increases for cases λ=µ and λ < µ and falls for the remaining case. λ µ∗ E(L) E(W) TC∗ Elapsed Time 0.3 1.3629 1.8445 6.1484 47.7613 291.0614 0.4 1.4473 2.3228 5.8070 48.1851 329.2360 0.5 1.5355 2.9194 5.8388 50.3765 321.3652 0.6 1.6241 3.6782 6.1304 53.9342 350.7149 0.8 1.7941 5.9485 7.4356 65.3713 359.1852 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 324 https://internationalpubls.com References [1] S.I. Ammar, A.A. El-Sherbiny, S.A. El-Shehawy and R.O. 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