Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 197 https://internationalpubls.com The Markovian Batch Arrival Queue with Differentiated Vacation Ms. K. Vijaya1*, Dr. B. Deepa2* 1*Research Scholar, Karpagam Academy of Higher Education,Coimbatore.email: vijayakirubai@gmail.com 2*Department of Mathematics, Faculty of Engineering, Karpagam Academy of Higher Education, Coimbatore-21, Tamil Nadu, India.email: Deepa09pragu@gmail.com Article History: Received: 10-04-2024 Revised: 29-05-2024 Accepted: 15-06-2024 Abstract: A Poisson Batch Arrival Queueing Model with a single server taking two different vacations at various rates is being considered for the study. The probability generating function method of the different states is considered for deriving the various performance measures of the states; numerical cases and cost studies are also extended. Keywords: Poisson batch arrival queue; differentiated vacations; probability generating function; 1. Introduction Queueing models where the server goes on vacation have applications in many areas, such as production and manufacturing systems, telecommunications, and service industries. In real time, during a normal time, if a customer is in the queue, the server is working at the normal service rate. When there are no customers in the system, the server leaves for a primary vacation for a certain period of time. Upon returning from a primary vacation, if the server finds no customers in the system, server leaves for another secondary vacation. After returning from the secondary vacation, the server will continue the service when customers are in the system. If there is no customers in the system, the server will wait for the arrival of the customers. Remarkable research work about server vacations on different vacations can be found in the queueing model literatures. Since Levy and Yechiali's [1] publication, where the concept was first introduced, many academics have become interested in server vacation queuing systems. Doshi [2] did a number of great surveys on those vacation models. Many scholars spent time on the working vacation concept, and different authors have modified the original model. Baba [3] utilized the matrix analytical method for examining a G1/M/1 working queue vacations. V.M. Chandrasekaran [4] carried out research on working vacation queueing models. ANFIS and cost optimization for a Markovian queue with operation vacation were examined by Sonali Thakur [5]. Utilizing matrix-geometric and ANFIS methods, computational findings are offered to assess the depth of the adaptive neural fuzzy inference system. The fundamental ideas of queueing theory were described by Binary kumar [6]. A bulk arrival poisson with vacation was the focus of research by Borthakur, A. Choudhury [7], G.Madan and Gautam Choudhury[8]A two phase bulk arrival queueing with vacation under Bernoulli schedule was introduced by K.C.Madan. Oliver C. Ibe, [9] considered it as a multiple vacation line system with separated vacations. J.Li,W.Liu [10] , initiated steady state analysis of a discrete bulk arrival with operating vacations.A.D.Banik [11] modelled G1/M/1/N queue with various operating vacations.B.Deepa and K.Kalidass [12] viewed a single server queueing model with operating Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 198 https://internationalpubls.com breakdowns and vacations .K.V.Vijayashree[13] considered a single server queueing line model and separated vacation and disturbance .Zhang.M [14] searched the appearance of only one server queue with operating vacations and vacation disturbance. Ganesh Sapkota [15] modelled the transient state of the M/M/C queueing line model with a standby server and reneging customers. Ammar, S. I [16] modelled transient state of a one server vacation queue with a waiting server and restless customers. Gupta [17] deals with two server model with operating vacation and reneging of customers due to impatience.V. Vijayalakshmi, B.Deepa and K.Kalidass[18] modelled cost of single server queue line with operating breakdowns using two – phase services. The layout of the paper is described as follows, the model considered here is presented in full detail in Section 2. The analysis in steady-state of the proposed model is explained in Section 3. The reliability measures and performance measures are obtained in Sections 4 and 5. Some illustrative numerical examples to point out the effects of the server are presented in Section 6. A cost optimization problem is discussed in section 7 and a few concluding remarks are given in Section 8. 2. The Model: Here a fixed batch-sized Markovian model with two non - identical vacation schemes is being considered. The arrival unit follows the distribution is Poisson with an arrival rate Ξ» and server is serving the arrival unit comes the distribution of exponentially distributed independent and identically distributed random variable with a rate Β΅. The arrival flow of customers enters the system in the way of batches/bulks. The capacity of batches is a random variable X of positive or neutral values with a probability distribution β„Žπ‘– = π‘ƒπ‘Ÿ{𝑋 = 𝑖} , 𝑖 β‰₯ 1. When there are zero customers in the system, the server leaves for a primary vacation at a rate π‘Ÿ1 for a certain period of time. Upon return from a primary vacation, if the server finds zero customers in the system, the server leaves for another vacation named as a secondary vacation at rate π‘Ÿ2. However, the server is from the secondary vacation then continue the service in normal state, and it will not go for vacation again. The arrival, service, and vacation periods are independent random variables of each other. Let B(t) be state of the server at t(time), 𝐡(𝑑) = { 0, π‘‘β„Žπ‘’ π‘ π‘’π‘Ÿπ‘£π‘’π‘Ÿ 𝑖𝑛 π‘›π‘œπ‘Ÿπ‘šπ‘Žπ‘™ 1, π‘‘β„Žπ‘’ π‘ π‘’π‘Ÿπ‘£π‘’π‘Ÿ 𝑖𝑛 π‘π‘Ÿπ‘–π‘šπ‘Žπ‘Ÿπ‘¦ π‘£π‘Žπ‘π‘Žπ‘‘π‘–π‘œπ‘› π‘ π‘‘π‘Žπ‘‘π‘’ 2, π‘‘β„Žπ‘’ π‘ π‘’π‘Ÿπ‘£π‘’π‘Ÿ 𝑖𝑛 π‘ π‘’π‘π‘œπ‘›π‘‘π‘Žπ‘Ÿπ‘¦ π‘£π‘Žπ‘π‘Žπ‘‘π‘–π‘œπ‘› π‘ π‘‘π‘Žπ‘‘π‘’ Let X(t) be the count of consumers attending in the system at t(time). Then {(𝐡(𝑑), 𝑋(𝑑)), 𝑑 β‰₯ 0} is considered a Continuous time MARKOV CHAIN. Let 𝑃𝑖,𝑛(𝑑) = π‘ƒπ‘Ÿπ‘œπ‘{𝐡(𝑑) = 𝑖 , 𝑋(𝑑) = 𝑛} , 𝑛 β‰₯ 0,𝑖 = 0,1,2 3. Steady State Analysis The Steady State governing equations are derived as πœ†π‘ƒ2,0 = π‘Ÿ1𝑃1,0 , 𝑛 = 0 _________(1) (π‘Ÿ2 + πœ†)𝑃2,𝑛 = πœ† βˆ‘ β„Žπ‘– 𝑃2,π‘›βˆ’1 , 𝑛 β‰₯ 1 ________(2) 𝑛 𝑖=1 ∴ ( πœ† + πœ‡)𝑃0,1 = πœ‡π‘ƒ0,2 + π‘Ÿ1𝑃1,1 + π‘Ÿ2𝑃2,1 , 𝑛 = 1 ________(3) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 199 https://internationalpubls.com ∴ ( πœ† + πœ‡)𝑃0,𝑛 = πœ‡π‘ƒ0,𝑛+1 + π‘Ÿ1𝑃1,𝑛 + π‘Ÿ2𝑃2,𝑛 + πœ† βˆ‘ β„Žπ‘– 𝑃0,π‘›βˆ’1, 𝑛 𝑖=1 𝑛 β‰₯ 2 _______(4) (πœ† + π‘Ÿ1)𝑃1,0 = πœ‡π‘ƒ0,1 , 𝑛 = 0 ________(5) (πœ† + π‘Ÿ1)𝑃1,𝑛 = πœ† βˆ‘ β„Žπ‘– 𝑃1,π‘›βˆ’1, 𝑛 β‰₯ 1 __________(6) 𝑛 𝑖=1 Probability generating function is βˆ‘ 𝑃2,𝑛𝑧𝑛 = 𝑃2(𝑧), ∞ 𝑛=0 βˆ‘ 𝑃0,𝑛𝑧𝑛 = 𝑃0(𝑧) ∞ 𝑛=1 , βˆ‘ 𝑃1,𝑛𝑧𝑛 = 𝑃1(𝑧)∞ 𝑛=0 𝐻(𝑧) = βˆ‘ β„Žπ‘›π‘§π‘›βˆž 𝑛=1 (πœ† + π‘Ÿ2)𝑃2(𝑧) = π‘Ÿ2𝑃2,0 + π‘Ÿ1𝑃1,0 + πœ†π»(𝑧)𝑃2(𝑧) _________(7) 𝑃2(𝑧) = π‘Ÿ1𝑃1,0 + π‘Ÿ2𝑃2,0 πœ†(1 βˆ’ 𝐻(𝑧)) + π‘Ÿ2 ________(8) 𝑃2(1) = π‘Ÿ1𝑃1,0 + π‘Ÿ2𝑃2,0 π‘Ÿ2 __________(9) πœ†π‘ƒ0(𝑧) + πœ‡π‘ƒ0 (𝑧) = βˆ’πœ‡π‘ƒ0,1 + 1 2 πœ‡π‘ƒ0(𝑧) βˆ’ π‘Ÿ1𝑃1,0 + π‘Ÿ1𝑃1(𝑧) + πœ†π»(𝑧)𝑃0(𝑧) βˆ’ π‘Ÿ2𝑃2,0 + π‘Ÿ2𝑃2(𝑧) πœ†π‘ƒ0(𝑧) βˆ’ πœ†π»(𝑧)𝑃0(𝑧) βˆ’ 1 2 πœ‡π‘ƒ0(𝑧) + πœ‡π‘ƒ0 (𝑧) βˆ’ π‘Ÿ1𝑃1(𝑧) βˆ’ π‘Ÿ2𝑃2(𝑧) = βˆ’πœ‡π‘ƒ0,1 βˆ’ π‘Ÿ1𝑃1,0 βˆ’ π‘Ÿ2𝑃2,0 (πœ†(1 βˆ’ πœ†π»(𝑧))𝑃0(𝑧) + πœ‡ (1 βˆ’ 1 𝑧 ) 𝑃0(𝑧)) βˆ’ π‘Ÿ1𝑃1(𝑧) βˆ’ π‘Ÿ2𝑃2(𝑧) = βˆ’πœ‡π‘ƒ0,1 βˆ’ π‘Ÿ1𝑃1,0 βˆ’ π‘Ÿ2𝑃2,0 [πœ† (1 βˆ’ πœ†π»(𝑧)) + πœ‡(1 βˆ’ 1 𝑧 )] 𝑃0(𝑧) βˆ’ π‘Ÿ1𝑃1(𝑧) βˆ’ π‘Ÿ2𝑃2(𝑧) = βˆ’πœ‡π‘ƒ0,1 βˆ’ π‘Ÿ1𝑃1,0 βˆ’ π‘Ÿ2𝑃2,0______(10) [πœ†(1 βˆ’ 𝐻(𝑧)) + π‘Ÿ1]𝑃1(𝑧) = πœ‡π‘ƒ0,1 ________(11) ∴ 𝑃1(𝑧) = πœ‡π‘ƒ0,1 πœ†(1 βˆ’ πœ†π»(𝑧)) + π‘Ÿ1 __________(12) 𝑃1(1) = πœ‡π‘ƒ0,1 π‘Ÿ1 From (10), 𝑃0(𝑧) = π‘Ÿ1𝑃1(𝑧) + π‘Ÿ2𝑃2(𝑧) βˆ’ πœ‡π‘ƒ0,1 βˆ’ π‘Ÿ1𝑃1,0 βˆ’ π‘Ÿ2𝑃2,0 (1 βˆ’ πœ†π»(𝑧)) + πœ‡(1 βˆ’ 1 𝑧 ) 𝑃0(𝑧) = (βˆ’πœ‡π‘ƒ0,1 ) [πœ†2(1 βˆ’ 𝐻(𝑧)) 2 + π‘Ÿ2 πœ†(1 βˆ’ πœ†π»(𝑧))] + [βˆ’π‘Ÿ1𝑃1,0 βˆ’ π‘Ÿ2𝑃2,0] [πœ†2(1 βˆ’ 𝐻(𝑧)) 2 + π‘Ÿ1 (1 βˆ’ 𝐻(𝑧))] πœ†3(1 βˆ’ 𝐻(𝑧)) 3 + (π‘Ÿ1 + π‘Ÿ2)πœ†2(1 βˆ’ 𝐻(𝑧)) 2 + π‘Ÿ1 π‘Ÿ2 (1 βˆ’ 𝐻(𝑧)) +πœ‡ (1 βˆ’ 1 𝑧 ) πœ†2(1 βˆ’ 𝐻(𝑧)) +(π‘Ÿ1 + π‘Ÿ2)πœ‡ (1 βˆ’ 1 𝑧 ) πœ†(1 βˆ’ 𝐻(𝑧)) + π‘Ÿ1 π‘Ÿ2πœ‡ (1 βˆ’ 1 𝑧 ) _______(13) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 200 https://internationalpubls.com When z=1 𝑃0(1) = 0 0 Using L’Hospitals rule, we get P0(z) = (βˆ’ΞΌP0,1 )[βˆ’2Ξ»2(1 βˆ’ H(z))Hβ€²(Z) βˆ’ r2 Ξ»Hβ€²(z)] + [βˆ’r1P1,0 βˆ’ r2P2,0][βˆ’2Ξ»21 βˆ’ H(z))Hβ€²(Z) βˆ’ r1Ξ»Hβ€²(z) ] βˆ’ ΞΈ1 βˆ’3Ξ»3(1 βˆ’ H(z)) 2 Hβ€²(z) βˆ’ (r1 + r2)2Ξ»21 βˆ’ H(z))Hβ€²(Z) βˆ’ r1 r2 Hβ€²(z) +ΞΌ 1 z2 Ξ»2(1 βˆ’ H(z)) 2 βˆ’ 2 (1 βˆ’ 1 z ) ΞΌΞ»2(1 βˆ’ H(z))Hβ€²(z) +(r1 + r2)ΞΌΞ» 1 z2 (1 βˆ’ H(z)) βˆ’ (r1 + r2)ΞΌ (1 βˆ’ 1 z ) Ξ»Hβ€²(z) + r1 r2ΞΌ 1 z2 βˆ’ ΞΈ2 __________(14) 𝑃0(1) = πœ‡π‘Ÿ2 πœ†π»β€²(1)𝑃0,1 + π‘Ÿ1 2 πœ†π»β€²(1)𝑃1,0 + π‘Ÿ1 π‘Ÿ2πœ†π»β€²(1)𝑃2,0 π‘Ÿ1 π‘Ÿ2(πœ‡ βˆ’ πœ†π»β€²(1)) = 𝑄1(1) 𝑄2(1) ________(15) 𝑃0β€²(𝑧) = 𝑄2𝑄1β€² βˆ’ 𝑄1𝑄2β€² 𝑄2 2 𝑃0β€²(1) = 𝑄2(1)𝑄1β€²(1) βˆ’ 𝑄1(1)𝑄2β€²(1) {𝑄2(1)}2 𝑄1 β€² (𝑧) = βˆ’πœ‡π‘ƒ0,1 [βˆ’2πœ†2(1 βˆ’ πœ†π»(𝑧))𝐻′′(𝑧) + 2πœ†2𝐻′(𝑧) βˆ’ π‘Ÿ2πœ†π»β€²β€²(𝑧)] + [βˆ’π‘Ÿ1𝑃1,0 βˆ’ π‘Ÿ2𝑃2,0][βˆ’2πœ†2(1 βˆ’ πœ†π»(𝑧))𝐻′′(𝑧) + 2πœ†2𝐻′(𝑧)2 βˆ’ π‘Ÿ1πœ†π»β€²β€²(𝑧)] 𝑄1 β€² (1) = βˆ’πœ‡π‘ƒ0,1 [2πœ†2𝐻′(1)2 βˆ’ π‘Ÿ2πœ†π»β€²β€²(1)] βˆ’ [π‘Ÿ1𝑃1,0 + π‘Ÿ2𝑃2,0][2πœ†2𝐻′(1)2 βˆ’ π‘Ÿ1πœ†π»β€²β€²(1)] 𝑄2 β€² (1) = 2πœ†2(π‘Ÿ1 + π‘Ÿ2)𝐻′(1)2 βˆ’ π‘Ÿ1 π‘Ÿ2πœ†π»β€²β€²(1) βˆ’ 2(π‘Ÿ1 + π‘Ÿ2 )πœ†π»β€²(1) βˆ’ 2π‘Ÿ1 π‘Ÿ2πœ‡ 𝑃0 β€²(1) = 𝑄 + 𝑅 + 𝑆𝑇 π‘ˆ _______________(16) Q=π‘Ÿ1 π‘Ÿ2πœ‡ βˆ’ π‘Ÿ1 π‘Ÿ2πœ†π»β€²(1) βˆ’ πœ‡π‘ƒ0,1 (2πœ†2𝐻′(1)2 βˆ’ π‘Ÿ2πœ†π»β€²β€²(1)) R=[βˆ’π‘Ÿ1𝑃1,0 βˆ’ π‘Ÿ2𝑃2,0]2πœ†2𝐻′(1)2 βˆ’ π‘Ÿ1πœ†π»β€²β€²(1) βˆ’ (πœ‡π‘Ÿ2 πœ†π»β€²(1)𝑃0,1 S=π‘Ÿ1 2 πœ†π»β€²(1)𝑃1,0 + π‘Ÿ1 π‘Ÿ2πœ†π»β€²(1)𝑃2,0 T=[2πœ†2(π‘Ÿ1 + π‘Ÿ2)𝐻′(1)2 βˆ’ π‘Ÿ1 π‘Ÿ2πœ†π»β€²β€²(1) βˆ’ 2(π‘Ÿ1 + π‘Ÿ2 )πœ†π»β€²(1) βˆ’ 2π‘Ÿ1 π‘Ÿ2πœ‡] U=π‘Ÿ1 2 π‘Ÿ2 2(πœ‡ βˆ’ πœ†π»β€²(1)) 2 We know 𝑃0(1) + 𝑃1(1) + 𝑃2(1) = 1 π‘Ÿ1 2πœ‡π‘ƒ1,0 + π‘Ÿ1 π‘Ÿ2πœ‡π‘ƒ2,0 + π‘Ÿ2πœ‡2𝑃0,1 = π‘Ÿ1 π‘Ÿ2(πœ‡ βˆ’ πœ†π»β€²(1))_____________(17) From (1) 𝑃2,0 = π‘Ÿ1 πœ† 𝑃1,0 ______________(18) From (5) 𝑃0,1 = πœ† + π‘Ÿ1 πœ‡ 𝑃1,0 _____________(19) Sub (18) & (19) in (17) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 201 https://internationalpubls.com 𝑃1,0 = π‘Ÿ1 π‘Ÿ2πœ†[πœ‡ βˆ’ πœ†π»β€²(1)] πœ‡[π‘Ÿ1 2 πœ† + π‘Ÿ1 2 π‘Ÿ2 2 + πœ†2π‘Ÿ2 + π‘Ÿ1 π‘Ÿ2πœ†] _______________(20) 𝑃0,1 = (πœ† + π‘Ÿ1)πœŒπ‘Ÿ1 π‘Ÿ2(πœ‡ βˆ’ πœ†π»(1)) πœ‡[ πœ†π‘Ÿ1 2 + π‘Ÿ1 π‘Ÿ2 2 + πœ†2π‘Ÿ2 + π‘Ÿ1 π‘Ÿ2πœ†] ______________(21) equation (16) 𝑃0β€²(1) and 𝑃1β€²(𝑧) = πœ‡π‘ƒ0,1[πœ†(1 βˆ’ 𝐻(𝑧)) + π‘Ÿ1] βˆ’2 𝐻′(𝑧) 𝑃1 β€²(1) = πœ‡π‘ƒ0,1𝐻′(1) π‘Ÿ1 2 and 𝑃2 β€²(𝑧) = [π‘Ÿ1𝑃1,0 + π‘Ÿ2𝑃2,0][πœ†(1 βˆ’ 𝐻(𝑧)) + π‘Ÿ2] βˆ’2 𝐻′(𝑧) 𝑃2 β€²(1) = π‘Ÿ1𝑃1,0 + π‘Ÿ2𝑃2,0 π‘Ÿ2 2 𝐻′(1) Now, customers count in the system=𝐸(𝑋) = 𝑃0 β€²(1) + 𝑃1 β€²(1) + 𝑃2 β€²(1) 4. Reliability Measures: Here, we discuss few reliability pointers of this model under deliberation. Specifically, we examine the accessibility of the server, the failure of the server. Accessibility is probability of the system operating rightly when it is asked to use it and server failure rate is probability of system will not work correctly. Let P(t) be probability of the server either serving to a customer or idle or on vacation. Steady state availability defined by 𝜌 < 1.Figure 1 illustrate the availability of the server against service rate πœ‡. We observe that the availability of the server under steady state decreases with increase in the arrival rate πœ† for the three different values of πœ† = 2, 2.1 and 2.2 as expected. Figure 2. depicts variation of the failure frequency of the server against the arrival rate πœ† for three different values of the primary vacation rate π‘Ÿ1=0.1,0.12 and 0.13. As we expect the failure frequency of the server increases with increasing in the arrival rate πœ†. The server accessibility is, Availability A= P0(1) it is shown in Figure 1. The server failure frequency is, Failure F=P1(1) +P2(1) it is shown in Figure 2. Fig.1 & 2-Availability and Failure of server Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 202 https://internationalpubls.com 5. Performance measure: Various measures of system performance can be developed from the steady state distribution. 1. Expected customers count in system 𝐸(𝑋) = 𝑃0 β€²(1) + 𝑃1 β€²(1) + 𝑃2 β€²(1) = βˆ‘ 𝑛 ∞ 𝑛=0 𝑃𝑛 2. Expected customers count in queue: 𝐸(𝑄) = βˆ‘ (𝑛 βˆ’ 1)∞ 𝑛=1 𝑃𝑛 3. Expected waiting time of a clients in system 𝐸(π‘Šπ‘ ) = 𝐸(𝑋) πœ† 4. Expected waiting t (time) of a customer in queue 𝐸(π‘Šπ‘ž) = 𝐸(𝑄) πœ† 5. The fraction of t(time), the server is in normal use 𝑄0 = 𝑃0( 1 ) 6. The fraction of t(time), the server in primary vacation 𝑄1 = 𝑃1( 1 ) 7. The fraction of t(time), server being secondary vacation 𝑄2 = 𝑃2(1) 6. Numerical Analysis: Here is the numeric expression for different performance indices that are provided. Figure 3 is a representation of the relationship between the server's service rate πœ‡ and arrival rate πœ† . Figure 4 is a representation of the relationship between the service rate πœ‡, the primary vacation rate π‘Ÿ1, and the arrival rate πœ†. Figure 5 is a representation of the relationship between the service rate πœ‡, secondary vacation rate π‘Ÿ2, and arrival rate πœ†. The picture 6 is for π‘Ÿ1= 0.1, π‘Ÿ2 = 0.2, and 𝐻’ (1) = 0.1. Here, the value 𝑃0,1 decreases as πœ‡ increases. It shows the effect of the service rate πœ‡ on the steady-state probability 𝑃0,1. We can see that all three probability curves for 𝑃0,1decrease as the probability increases the value of service rate πœ‡. We also see a rapid decline Probability curve 𝑃0,1for πœ† = 0.1, probability curve decreases from 𝑃0,1. It is slower for πœ† = 0.2 and πœ† = 0.4 when π‘Ÿ1= 0.1, π‘Ÿ2= 0.2, 𝐻’ (1) = 0.1. The figure 7 is for π‘Ÿ2=0.2 πœ† =0.3, 𝐻’ (1) =0.1. It was observed that the probability𝑃0,1decreases when the arrival rate increases and πœ‡ increases. The figure 8 is for π‘Ÿ1= 0.5, πœ† = 0.3 πœ† =0.3 = 0.1. Observed that as 𝑃0,1decreases, πœ‡ increases. Fig.3: Correlation between P0,1 verses πœ† & πœ‡ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 203 https://internationalpubls.com Fig.4: Correlation between P0,1 verses πœ† & π‘Ÿ1 Fig.5: Correlation between P0,1 verses πœ† & π‘Ÿ2 Fig.6: Correlation between P0,1 and service rate πœ‡ based on πœ† Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 204 https://internationalpubls.com Fig.7: Correlation between P0,1 and service rate πœ‡ based on π‘Ÿ1 Fig.8: Correlation between P0,1 and service rate πœ‡ based on π‘Ÿ2 Figure 9 is the representation of the relationship between primary vacation P1,0 with vacation rate π‘Ÿ1 and arrival rate πœ†. Fig.10 says the correlation between P1,0 and arrival rate πœ† when π‘Ÿ1=0.7, π‘Ÿ2=0.5, 𝐻’ (1) = 0.1 Figure 10 says, P1,0 decreases πœ† increases for different values of πœ‡ =4,7,20. Below figure 11 is for π‘Ÿ2=0.5, πœ‡ =6, 𝐻’ (1) =0.1. Here P1,0 value decreases the value of πœ† increases for different values of primary vacation rate π‘Ÿ1.It is observed that P1,0 value decreases the value of πœ† increases for different values of vacation rate π‘Ÿ1. It can be seen that all three probability curves for P1,0 decrease in probability as the value of the arrival rate πœ† increases. Also, when π‘Ÿ1= 1, the probability curve P1,0 is decreasing rapidly, and the probability curve is decreasing from P1,0. It is slow for π‘Ÿ1= 0.5 and π‘Ÿ1= 0.7. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 205 https://internationalpubls.com Fig.9: Correlation between P1,0 and arrival rate πœ† based on π‘Ÿ1 Fig.10: Correlation between P1,0 and arrival rate πœ† based on πœ‡ Fig.11: Correlation between P1,0 verses πœ† & π‘Ÿ1 Figure 12 is the representation of the relationship between secondary vacation P2,0 with vacation rate π‘Ÿ2 and arrival rate πœ† .The figure 13 is for π‘Ÿ2=0.5, πœ‡=2, 𝐻’ (1) =0.1. It is noticed that πœ† increases P2,0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 206 https://internationalpubls.com decreases. Figure13 is the representation of the Correlation between P2,0 and arrival rate πœ† based on π‘Ÿ1 values. Figure 14 says, P2,0 decreases when πœ† increases. It is noticed that P2,0 decreases when πœ† increases for different values of π‘Ÿ2 when π‘Ÿ1=2, πœ‡ =2, 𝐻’ (1) =0.1. We see that as the value of the arrival rate πœ† increases, the probabilities of all three probability curves for P2,0 decrease. Even when π‘Ÿ2= 0.2, the probability curve P2,0 decreases rapidly, and from P2,0 the probability curve decreases. It is slow for π‘Ÿ2= 0.4 and π‘Ÿ2= 0.6. Fig.12: Correlation between P2,0 verses πœ† & π‘Ÿ2 Fig.13 Correlation of P2,0 and arrival rate πœ† based on π‘Ÿ1 Fig.14: Correlation between P2,0 and arrival rate πœ† based on π‘Ÿ2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 207 https://internationalpubls.com Fig.15: Correlation between E(X) (customers count in the system) and arrival rate πœ† Figure 15 says the correlation between E(X) and arrival rate πœ† for πœ‡ = .08, .09, .1 Here E(X) increasing with growing of πœ† 7.1 COST - BENEFIT ANALYSIS: We have created an anticipated cost function F / unit time for Markovian vacation queue and batch arrival size. Our aim is to compute the optimal values for πœ‡ to minimize function cost. Describe the cost 𝐢𝑖 , 𝑖 = 1 π‘‘π‘œ 6 elements as follows In this sub section, we have developed an anticipated cost function F to find the optimum normal πœ‡βˆ— and total expected optimum cost, 𝐹(πœ‡βˆ—).We define the cost elements are C1= service cost / unit time when the server is on C2= holding cost /unit time when the server is on C3= cost / unit time when a customer connects queue in normal time C4= cost / unit time when a customer connects queue in primary vacation time C5= cost /unit time when a customer connects the queue in secondary vacation time C6= cost / unit time when a customer waiting time in the system Based on the interpretations of every cost element and the system performance metrics, an expected total cost function / unit time obtained 𝐹(πœ‡βˆ—) = 𝐢1πœ‡ + 𝐢2𝐸(𝑋) + 𝐢3𝑃0,1 + 𝐢4𝑃1,0 + 𝐢5𝑃2,0 + 𝐢6𝐸(π‘Šπ‘ ) The aim is calculating the optimal rate πœ‡βˆ— to minimize the function cost F. We have followed to calculate the convexity of the total cost by the method of direct search 7.2 Direct search method The following are the cost parameter by using assumptions 𝐢1 = 5, 𝐢2 = 15, 𝐢3 = 3, 𝐢4 = 6, 𝐢5 = 4, 𝐢6 = 10 Here the Numerical examples to define the πœ‡ π‘£π‘Žπ‘™π‘’π‘’ using direct search method. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 208 https://internationalpubls.com From figure16, cost function and tables here that for different values of πœ† = 0.5,0.8,1 found 75.3865, 70.51219, 78.25883 these are minimum expected cost. The figure 17 of cost output and tables shows that for different values of π‘Ÿ1 = 0.5,0.6,0.7 received 132.707, 112.815, 96.3627 which are minimum expected cost. From figure18, cost function and tables here that for different values of π‘Ÿ2 = 1,3,7 found 84.53188, 49.66225, 49.66225 these are minimum expected cost. Fig.16: Correlation between the cost F and service rate πœ‡ based on πœ† Figure 16. The result of πœ‡ on the cost function F. Convexity of cost graph is exhibited, Minimum cost is found for this queue model when the different values of arrival rate πœ†. π‘Ÿ2 = 1.2, π‘Ÿ1 = 1, 𝐻′(1) = 1, 𝐻′′(1) = 0.3 and 𝐢1 = 5, 𝐢2 = 15, 𝐢3 = 3, 𝐢4 = 6, 𝐢5 = 4, 𝐢6 = 10.The table is 𝝁 πœ† = 0.5 πœ† = 0.8 πœ† = 1 𝑃1,0 𝐸(π‘Šπ‘ ) F-Cost 𝑃1,0 𝐸(π‘Šπ‘ ) F-Cost 𝑃1,0 𝐸(π‘Šπ‘ ) F-Cost 2.5 0.117834 2.185464 76.53421 0.132146 2.290377 78.8957 0.130909 3.186639 103.6493 3.5 0.090179 1.96252 75.3865 0.107081 1.83848 71.41864 0.111317 2.363622 85.83763 4.5 0.072737 1.838736 76.96695 0.088769 1.595472 69.69908 0.094276 1.965868 79.81138 5.5 0.060865 1.760273 79.79719 0.075484 1.444192 70.51219 0.081142 1.729574 78.25883 6.5 0.052293 1.706152 83.29956 0.065544 1.341025 72.65519 0.071006 1.572616 78.90495 7.5 0.045824 1.666583 87.20413 0.057868 1.266187 75.58179 0.06303 1.460662 80.79865 8.5 0.040773 1.6364 91.36823 0.051777 1.209424 79.00865 0.056622 1.376739 83.4696 Table1.The result of πœ‡ against the cost output for different points of πœ† Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 209 https://internationalpubls.com Fig.17: Correlation between the cost F and service rate πœ‡ based on π‘Ÿ1 Figure 17. The result of πœ‡ on the cost function F πœ† = 1. π‘Ÿ2 = 1, 𝐻′(1) = 1, 𝐻′′(1) = 0.5 and 𝐢1 = 5, 𝐢2 = 15, 𝐢3 = 3, 𝐢4 = 6, 𝐢5 = 4, 𝐢6 = 10 The table is 𝝁 π‘Ÿ1 = 0.5 π‘Ÿ1 = 0.6 π‘Ÿ1 = 0.7 𝑃1,0 𝐸(π‘Šπ‘ ) F-Cost 𝑃1,0 𝐸(π‘Šπ‘ ) F-Cost 𝑃1,0 𝐸(π‘Šπ‘ ) F-Cost 4.2 0.074697 7.164505 148.0841 0.068027 3.153055 122.706 0.060469 3.804248 100.9624 5.2 0.063958 6.344968 138.8077 0.058247 2.780227 115.9699 0.051775 3.370196 97.13739 6.2 0.055702 5.804327 134.3884 0.050728 2.53549 113.2647 0.045092 3.086235 96.3627 7.2 0.049247 5.420873 132.707 0.04485 2.362428 112.815 0.039866 2.885877 97.2875 8.2 0.044091 5.134748 132.7213 0.040155 2.233551 113.7571 0.035693 2.736904 99.25701 9.2 0.039892 4.913068 133.8582 0.03633 2.133845 115.617 0.032294 2.621781 101.9149 10.2 0.036411 4.736266 135.777 0.03316 2.054407 118.1151 0.029476 2.530143 105.0503 11.2 0.033482 4.591966 138.2621 0.030493 1.989626 121.0748 0.027105 2.455465 108.5308 Table2.The result of πœ‡ against the cost output for different points of π‘Ÿ1.Convexity of the cost graph is exhibited; Minimum cost is found for this queue model when the different values of primary vacation rate π‘Ÿ1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 210 https://internationalpubls.com Fig.18: Correlation between the cost F and service rate πœ‡ based on π‘Ÿ2 Figure 18. The result of πœ‡ on the cost function F. Convexity of cost graph is exhibited, Minimum cost is found for this queue model when the different values of secondary rate π‘Ÿ2. πœ† = 1, π‘Ÿ1 = 1, 𝐻′(1) = 1, 𝐻′′(1) = 0.5 and 𝐢1 = 5, 𝐢2 = 15, 𝐢3 = 3, 𝐢4 = 6, 𝐢5 = 4, 𝐢6 = 10 The table is 𝝁 π‘Ÿ2 = 1 π‘Ÿ2 = 3 π‘Ÿ2 = 7 𝑃1,0 𝐸(π‘Šπ‘ ) F-Cost 𝑃1,0 𝐸(π‘Šπ‘ ) F-Cost 𝑃1,0 𝐸(π‘Šπ‘ ) F-Cost 2.2 0.059846 4.639807 128.7307 0.106257 0.947738 64.46707 0.123967 0.274697 43.36624 3.2 0.051859 3.232333 98.84934 0.092076 0.622104 51.78099 0.107422 0.194398 39.26325 4.2 0.043788 2.619898 88.67432 0.077745 0.490917 49.66225 0.090703 0.165711 41.02152 5.2 0.037492 2.273158 85.08117 0.066568 0.41929 50.77505 0.077663 0.150908 44.35032 6.2 0.032653 2.049288 84.53188 0.057975 0.374001 53.31656 0.067638 0.141854 48.32851 7.2 0.028869 1.892604 85.64727 0.051257 0.342731 56.61855 0.059799 0.135738 52.63837 8.2 0.025847 1.776737 87.77416 0.045891 0.319825 60.37437 0.05354 0.131328 57.14065 9.2 0.023385 1.687546 90.56223 0.04152 0.302315 64.42307 0.04844 0.127996 61.76458 Table3.The result of πœ‡ against the cost output for different points of π‘Ÿ2 8. Conclusion: An 𝑀π‘₯/𝑀/1 queueing model with differentiated vacation is studied in this paper A probability generating function method is used to find the model's steady-state and steady-state probabilities. Further, the various system performance measures are also analyzed numerically and graphically. A cost function is also constructed to obtain the optimal (minimum) cost function corresponding to the optimal service rate through the direct search method. Mathematica is used for presenting 3Dimensional graphical pictures and MAT LAB software is also used for presenting the cost function of the above queueing model. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 211 https://internationalpubls.com References: [1] Y.Levy and U.Yechiali β€œUtilization of idle time in an M/G/1 queueing system”, Management science, Vol.22, no.2, pp.202-211,1975. [2] Doshi B.T., Queueing Systems with vacations- a survey, Queueing Systems, (1986),29-66. [3] Y.Baba, β€œAnalysis of a G1/M/1 queue with multiple working vacations”, Operations Research letters, Vol.33, no.2, pp 201-209, 2005. [4] V.M.Chandrasekaran, β€œA Survey on working vacation Queueing models”, International journal of Pure and applied Mathematics, Vol.106, no.6, pp 33-41, 2016. [5] Sonali Thakur β€œANFIS and cost optimization for Markovian Queue with operational vacation”, International journal of Mathematical, Engineering and Management Sciences, Vol 6, no.3, pp894-910, 2021. [6] Binary Kumar, β€œSome Basic concepts in queueing theory”, International advanced research journal in science, engineering and technology, vol 2, issue 12, December 2015. [7] Borthakur. A and Choudhury, G. [1997], on a batch arrival poisson queue with generalized vacation. Sankhya; The Indian journal of statistics 59 (3): 369-383. [8] Gautam Choudhury and urdan K.C (2004), A two phase batch arrival queueing system with a vacation time under Bernoulli schedule, applied mathematics and computation ,149, 337-349. [9] Oliver C. Ibe and olubukola A. Isijola , β€œM/M1 multiple vacation queueing systems with differentiated vacations β€œ, Modelling and simulation in engineering volume 2014, pp6 ,2014 [10] J.Li,W.Liu and N.Tian, β€œSteady-state analysis of a discrete time batch arrival queue with working vacations”, Performance Evaluation, Vol.67, no.10, pp.897-912, 2010 [11] A.D.Banik, U.C. Gupta and S.S. Pathak, β€œOn the G1/M/1/N queue with multiple working vacations-analytic analysis and computation”, Applied Mathematical Modelling, Vol.31, no.9, pp.1701-1710, 2007. [12] B.Deepa and K.Kalidass β€œAn M/M/1/N Queue with working Breakdowns and Vacations”, International Journal of Pure and Applied Mathematics Vol.119, no.10, 2018, pp.859-873. [13] K.V.Vijayshree, B.Janani and K.Ambine, β€œM/M/1 Queueing model with differentiated vacation and interruption”, Global and Stochastic Analysis; Vol.8, no.2, 2021. [14] Zhang.M and Hou.Z, β€œPerformance analysis of M/G/1 queue with working vacations and vacation interruption”, Journal of Computational and Applied Mathematics, Vol 234, pp.2977-2985, 2010. [15] Ganesh Sapkota and R.P.Ghimire, β€œMathematical Analysis of M/M/C Vacation Queueing model with a waiting server and Impatient Customers”, Journal of Mathematics and Statistics, Vol.18, pp. 36-48, 2022. [16] Ammar.S.I, β€œ Transient solution of an M/M/1 vacation queue with a waiting server and Impatient customers”, Journal of the Egyptian Mathematical Society, 25(3), pp.337-342, 2016. [17] S.Gupta, P.K. Joshi, K.N. Rajeshwari, β€œOptimization of M/M/2 Queueing Model with working vacations”, Vol 15, pp.31-41, 2022. [18] V.Vijayalakshmi, B.Deepa and K.Kalidass , Cost Analysis of M/M/1/N queue with working breakdowns and a two- phase service, Journal of physics, conference series,1850(2021),1742-6596.