Ratio Mathematica Volume 43, 2022 Analysis of Finite Population Stochastic modeling with State-Dependent Arrival and Service Facilities K Lakshmanan* S Padmasekaran† M Bhuvaneshwari‡ R Asokan§ Abstract This paper investigates a stock-dependent arrival process(SDAP) and queue- dependent service process(QDSP) in the stochastic queueing-inventory sys- tem(SQIS). The arriving units in the system generated from the finite source population. The arrival process holds the properties of quasi-random pro- cess and its intensity rate is defined based on the two-component demand rate(TCDR). The customers departure time is exponentially distributed. The concepts of non-SDAP and SDAP, non-QDSP and QDSP are to be gener- alized. The inventory system may have the perishable quality of the prod- ucts. It adopts the (s, Q) reordering policy whenever the replenishment is required. Further, the join probability distribution of a Markov process is derived and necessary system performance measures are computed. The comparative discussion is presented to improve the quality of this model. Keywords: Stock-dependent arrival process; Non-stock dependent arrival process; Two component demand rate; Finite population; queue-dependent service process 2020 AMS subject classifications: 90C15, 60G07. 1 *Professor of Mathematics, Kuwait American School of Education, Salmiya, PC 22063, Kuwait; lakshmanlingam@kas.edu.kw †Corresponding Author; Assistant Professor of Mathematics, Periyar University, Salem, Tamilnadu, India; padmasekarans@periyaruniversity.ac.in. ‡P. G. Assistant in Mathematics, Government Higher Secondary School, Sathirakudi, Ramanathapu- ram, Tamilnadu, India; bhabhu1977@gmail.com. §Professor of Mathematics, Madurai Kamaraj University, Madurai, Tamilnadu, India; asokan.maths@mkuniversity.org. 1Received on February 9, 2022. Accepted on December 1, 2022. Published on December 30, 2022. doi: 7 K. Lakshmanan, S Padmasekaran, M Bhuvaneshwari, R Asokan 1 Introduction As the continuous increment in the population growth, the demand of the people such as foods, cloths and public service etc are also skyrocketed in this century. So the analysis of queueing-inventory management is inevitable in this current situation, especially in that food items related queueing-inventory systems. The results can be applied effectively in queue and inventory management systems and the optimum cost will increase the economy also. This model consists of a perishable inventory system with queue-dependent service rate and customers from a finite source. Due to the decay of the food items after some certain period, its must to reorder it at some fixed inventory level to avoid economical loss. The arrival rate of customer is dependent on the number of items available in the shops. Also service rate is dependent on the mood of server. The following illustration will give the exact model idea. In the shopping malls and theaters, some stores are used to sell snack items for the customers. The possibility of customer occurrence is dependent on inventory level and also service rate can be dependent on the queue length. Here the population size is finite, since the only possible customers to purchase the snacks are from theater or the mall. Though there are many shops are available, most of the customers are interested to purchase from the shop where many items are available comparing to the shops where less items are available. Here the customer arrival depends on the inventory level. When a small queue is formed in the shop, server may serve slowly, speaking with someone or using mobile phone. Also server would provide a quick service if the queue length is big, in order to decrease the customer loss. Though the formed queue is bigger, customer won,t leave the queue as the service rate is high. This realistic situation motivates the author to develop the proposed model. In this model, stocks are replenished according to (s, Q) policy. 2 Literature review As many researchers show their interest of research on the Stochastic Queueing Inventory modeling, this area has developed enormously in certain period. Queuing in- ventory systems with retrial is in-fusible in all walk of life. Reshmi and Jose studied the Queuing inventory model, considering perishable items and customer retrial on Reshmi and Jose [2019]. Periyasamy considered a finite population perishable inventory system where server is looking for the customers from the orbit to provide service after com- pleting service to each primary customer on Periyasamy [2017]. Berman and Sapna Berman and Sapna [2002] discussed the rate of optimal service with perishable inven- tory in which instantaneous reordering policy was assumed. Considering the negative 10.23755/rm.v42i0.715. ISSN: 1592-7415. eISSN: 2282-8214. ©K Lakshmanan et al.. This paper is published under the CC-BY licence agreement. 8 Analysis of Finite Population Stochastic modeling exponential rate for the life time of stocks, Kalpagam and Arivagam Kalpakam and Ari- varignan [1988] analyzed the (s, S) inventory system in which stock one is evicted from the inventory whenever the demand or failure of item occurs. Sangeetha investigated the production optimal control of production time of perishable inventory system with finite source in order to get the minimal total cost on N. Sangeetha and Arivarignan [2015]. Alfres introduced the concept of occurring demand rate depends upon the stock level in the inventory system on Alfares [2007] and determined the total cost by variable hold- ing cost assuming holding cost per unit item to be a monotonically increasing function of spending time in the storage. Diana Tom Varghese and Dhanya Shajin Varghese and Shajin [2018] studied the state dependent demand on the continuous review M/M/1/S inventory model. K. Venkata Subbaiah et al. K. Venkata Subbaiah and Satyanarayana [2004] developed the perishable inventory model with stock dependent demand rate. Rathod and Bhathawala Rathod and Bhathawala [2013] analyzed the inventory system with stock dependent demand having variable holding cost and shortages. The effect of demand rate depending on stock level was discussed through the proposed logistical growth model of Tsoularis Tsoularis [2014]. A shortage free inventory model with stock dependent demand was analyzed by Datta and Pal on Datta and Pal [1990]. Sudhir Ku- mar Sahu et al. Sudhir Kumar Sahu and Sahoo [2008] developed an inventory system with stock dependent demand rate and constant deterioration with the possibilities of partial or complete backlog and without it. Shib Sankar Sana Sana and Sankar [2010] proposed an EOQ model for the perishable inventory item with discount rate and the demand depending on stock level. Mandal Mandal and S. [1989] derived an inventory system with consumption rate depending on stock level. For analyzing the local area management, Falin and Artalejo Falin and Artalejo [1998] proposed a retrial queue with finite source customer. Shophia Lawrence et al. A. Shophia Lawrence and Arivarignan [2013] discussed the perishable queueing-inventory system with demands from finite homogeneous source. Attahiru Sule Alfaa and Sapna Isotupa ? discussed an M/PH/k retrial queue with the finite source. K. Jeganathan K. Je- ganathan and Vigneshwaran [2015] analyzed the perishable inventory system with the possibility of server interruption and the multiple server vacation and customer is pro- vided service only when customer level reaches to a particular N and no customer is left behind the system after service started. Jeganathan Jeganathan [2015] discussed finite source inventory system with an additional service for some customers which is called bonus service. Artalejo and Lopez-Herrero investigated retrial queue involving finite population with an BSDE approach on Artalejo and Lopez-Herrero [2012]. Sivakumar analyzed the perishable inventory system with retrial demand from finite source without service on Sivakumar [2009]. Shanthikumar and Yao Shanthikumar and Yao [1988] studied the upper and lower bounds on a closed queuing network with the queue dependent service rate. Menich ronald Ronald [1987] derived the optimal of shortest queue routing to the queue depen- 9 K. Lakshmanan, S Padmasekaran, M Bhuvaneshwari, R Asokan dent service station considering a general Markovian system. Avhishek Chatterjee et al. Avhishek Chatterjee and varshney lav [2017] studied the information-theoretic limit of reliable information processing using queue dependent service facility. Jeganathan et al.[2021] proposed a finite inventory single server system and analyzed the queue dependent service rate. Though the large number of researches have been done in this area, there is a re- search gap in analyzing stock dependent demand rate on the finite source queuing inven- tory system with the queue dependent service rate and retrial customer. As the demand rate depending upon stock level on the inventory and service rate depends on the queue length, this model simulates the realistic situation. 3 Model Developing 3.1 Mathematical Formulation of the model This model deals the state dependent arrival and queue dependent service processes in a single server Markovian queueing-inventory system(SSMQIS) in a finite source environment. The system holds maximum of S units of inventory product in its storage place. It allows the customers to buy the product from a finite source, N only. It admits the arriving customers into the finite waiting hall of size N. There is only two possible choice of a customer such that they must be either free or in the waiting hall at any time. The appearance of arrival process generates a output process called quasi-random process; that is, the probability that any particular customer generates a request for demand in any interval (t, t + dt) is θrdt + o( dt)(r ∈ BS0 ) as d t → 0 if the customer is free at time t and zero if the customer is in waiting hall at time t, independently of the behavior of any other customers. The arrival process of any individual customer is non- homogeneous, since the generation of arrivals must dependent upon the current stock level of the system. This non-homogeneous arrival streams come under the category of state dependent arrival stream. Next, the service pattern is processed following a first come and first serve(FCFS) service discipline. The service time of any customer at time t is non-homogeneous and exponentially distributed. That is, µs (s ∈ BN1 ) is the service rate of an individual at any time. This service process comes under the category of state dependent service processes. After each service completion, there will be one unit dropped in the storage place. The stored products in the system does not have any guarantee about its life time till it will be sold. It may have the deteriorating quality. So this deterioration process follows exponential distribution and have the intensity rate rα1 where r ∈ BS1 . The service and deterioration processes cause the depletion of an inventory product unit by unit. At one fine stage, the current number of product in the storage system will reach the predetermined value s. As and when the maximum inventory level reduced to s or 10 Analysis of Finite Population Stochastic modeling less than s, then the replenishment process will be triggered immediately. Each time there are Q = (S − s) items will be replaced whenever the reorder required. This policy is known as (s, Q) reordering policy and this processing time is exponentially distributed with an intensity rate α. The defined arrival and service rates are ordered, θ0 ≤ θ1 ≤ θ2 ≤ · · · ≤ θS and µ1 ≤ µ2 ≤ · · · ≤ µN as an increasing manner. When the case θ1 = θ2 = · · · = θS = θ, the considered model comes under the category of two component demand rate. That is the arrival rate is homogeneous in the positive stock period and during the stock out period, it is θ0. When the case µ1 = µ2 = · · · = µN = µ means that the service rate become homogeneous. Remark 3.1. • For a numerical computation θr can be defined by θr rβ1, 0 < β1 ≤ 1 and r ∈ BS1 . • For a numerical computation µs can be defined by µs sβ2, 0 ≤ β2 ≤ 1 and s ∈ BN1 . • The case β1 = 0 and β2 = 0 explores the result of non-stock dependent arrival process and non- queue dependent service process of the proposed model. 4 Analytical Discussion of the Model Let { (R1(t), R2(t)) ; t ≥ 0 } be a stochastic process having state space {(r1, r2) : r1 ∈ BS0 and r2 ∈ BN0 } satisfies the Markov process, where R1(t) denotes the level of inven- tory at time t and R2(t) denotes the number of customers in the orbit at time t. The transition from any state (r1, r2) to other state (r′1, r ′ 2) at any interval is denoted by P ((r1, r2) , (r ′ 1, r ′ 2)). Any y items in the inventory perish alone at the rate of r1γ and the occurrence of primary demand is (N − r2) θr1 from any one of the sources (N − r2). Hence, the probability of transition is P ((r1, r2) , (r1 − 1, r2)) = r1α1 r1 ∈ BS1 and r2 ∈ BK0 . Since the service rate is queue dependent service rate P ((r1, r2) , (r1 − 1, r2 − 1)) = µr2 where r1 ∈ BS1 and r2 ∈ BK1 . If the arrival rate is dependent on inventory, arriving customers enter into the waiting hall. So the probability of the transition from the state (r1, r2) to the state (r1, r2 + 1) is P ((r1, r2) , (r1, r2 + 1)) = (K − r2) θr1 where r1 ∈ BS0 , r2 ∈ B K−1 0 . When Q items are ordered, the probability of transition from the state (r1, r2) to state (r1 + Q, r2) for all r2 and r1 ∈ Bs0 is given by P ((r1, r2) , (r1 + Q, z)) = α. The rate of other transitions is zero. The sum of each row of this matrix should be zero. Hence, the diagonal entry is multiplied by a negative sign after summing all the entries from the row. All the possible transitions are given below. 11 K. Lakshmanan, S Padmasekaran, M Bhuvaneshwari, R Asokan R ((r1, r2) , (r ′ 1, r ′ 2)) =   r1α1, r ′ 1 = r1 − 1, r1 ∈ BS1 , r′2 = r2, r2 ∈ BK0 , µr2, r ′ 1 = r1 − 1, r1 ∈ BS1 , r′2 = r2 − 1, r2 ∈ BK1 , (K − r2)θr1, r′1 = r1, r1 ∈ BS0 , r′2 = r2 + 1, r1 ∈ B K−1 0 , α, r′1 = r1 + Q, r1 ∈ Bs0, r′2 = r2, r2 ∈ BK0 , −(δ̄K,r2(K − r2)θr1 + α), r′1 = r1, r1 ∈ B00, r′2 = r2, , r2 ∈ BK0 , −(δ̄K,r2(K − r2)θr1 + α + δ̄0,r2µr2 + r1α1), r′1 = r1, r1 ∈ Bs1, r′2 = r2, , r2 ∈ BK0 , −(δ̄K,r2(K − r2)θr1 + δ̄0,r2µr2 + r1α1), r′1 = r1, r1 ∈ BSs+1, r′2 = r2, , r2 ∈ BK0 , 0, Otherwise. The block partitioned matrices of the proposed model is structured as follows: R =   Ly, r ′ 1 = r1, r1 ∈ BS0 , My, r ′ 1 = r1 − 1, r1 ∈ BS1 , N, y′ = Q + r1 r1 ∈ Bs0, 0, Otherwise. For r1 ∈ BS1 , Mr1 =   r1α1, r ′ 2 = r2, r2 ∈ BK0 , , µr2 r ′ 2 = r2 − 1, r2 ∈ BK1 , 0, Otherwise. For r1 ∈ Bs0, N = { α, r′2 = r2, r2 ∈ BK0 , 0, Otherwise. For r1 = 0, Lr1 =   (K − r2)θ0, r′2 = r2 + 1, r2 ∈ B K−1 0 , −((K − r2)θ0 + α), r′2 = r2, r2 ∈ B K−1 0 , −α, r′2 = r2, r2 ∈ BKK , 0, Otherwise. 12 Analysis of Finite Population Stochastic modeling For r1 ∈ Bs1, Lr1 =   (K − r2)θr1, r′2 = r2 + 1, r2 ∈ B K−1 0 , −(δ̄K,r2(K − r2)θr1 + α + δ̄0,r2µr2 + r1α1), r′2 = r2, r2 ∈ BK0 , 0, Otherwise. For r1 ∈ BSs+1, Lr1 =   (K − r2)θr1, r′2 = r2 + 1, r2 ∈ B K−1 0 , −(δ̄K,r2(K − r2)θr1 + δ̄0,r2µr2 + r1α1), r′2 = r2, r2 ∈ BK0 , 0, Otherwise. 4.1 Steady state analysis The structure of the homogeneous Markov process {(R1(t), R2(t); t ≥ 0} with finite state space indicates that it is irreducible. Hence, the limiting distribution is ξ(r1,r2) = lim t→∞ Pr {(R1(t) = r1, R2(t) = r2)|(R1(0), R2(0))} Let ξ = (ξ(0), ξ(1), . . . , ξ(S)) where each ξ(r1) = (ξ(r1,0), ξ(r1,1), . . . , ξ(r1,K)) for r1 ∈ BS0 which satisfies ξP = 0 and ξe = 1 (1) From the above we get the following equation ξ(r1)Lr1 + ξ (r1+1)Mr1+1 = 0 r1 ∈ B Q−1 0 , (2) ξ(r1)Lr1 + ξ (r1+1)Mr1+1 + ξ (r1−Q)N = 0 r1 = Q (3) ξ(r1)Lr1 + ξ (r1+1)Mr1+1 + ξ (r1−Q)N = 0 r1 ∈ BS−1Q+1, (4) ξ(r1)Lr1 + ξ (r1−Q)N = 0 r1 = S (5) Except the r1 = Q case, solving other equations recursively, we get, ξ(r1) = ξ(Q)∆r1, r1 ∈ B S 0 , where ∆i =   (−1)(Q−r1)(MQMQ−1 . . . Mr1+1)(L −1 Q−1L −1 Q−2 . . . L −1 r1 ), r1 ∈ B Q−1 0 , I r1Q, (−1)2Q+1−r1 C−r1∑ j=0 (MQMQ−1 . . . Ms+1−j)(L −1 Q−1L −1 Q−2 . . . L −1 s−j)NL −1 S−j (MS−jMS−j−1 . . . Mr1+1)(L −1 S−j−1L −1 S−j−2 . . . L −1 r1 ) r1 ∈ BSQ+1, 13 K. Lakshmanan, S Padmasekaran, M Bhuvaneshwari, R Asokan ξ(Q) can be yield solving ξ(Q) [ (−1)2Q+1−r1 S−r1∑ j=0 [ (MQL −1 Q−1MQ−1 . . . Ms+1−jL −1 s−j)NL −1 S−j(MS−jL −1 S−j−1MS−j−1 . . . . . . MQ+2L −1 Q+1) ] MQ+1 + LQ + (−1)QMQL−1Q−1MQ−1 . . . M1L −1 0 N ] = 0 ξ(Q) [ S∑ r1=Q+1 ( (−1)2Q−r1+1 S−r1∑ j=0 [ (MQL −1 Q−1MQ−1 . . . Ms+1−jL −1 s−j)NL −1 S−j(MS−jL −1 S−j−1 MS−j−1 . . . Mr1+1L −1 r1 ) ] MQ+1 ) + Q−1∑ r1=0 ( (−1)Q−r1MQL−1Q−1 . . . Mr1+1L −1 r1 ) +I ] e = 1. 5 System Performance Measures To make a detailed investigation of the proposed model, some significant system characteristics are to be computed as follows: 1. Expected present stock level E[psl] = ∑S r1=1 ∑K r2=0 r1ξ (r1,r2). 2. Expected reorder level E[reorder] = ∑K r2=1 µr2ξ (s+1,r2)+ ∑K r2=0 (s+1)α1ξ ((s+1),r2). 3. Expected perishable rate E[perishable] = ∑S r1=1 ∑K r2=0 r1α1ξ (r1,r2). 4. Expected number of customers in the waiting hall E[CWH] = ∑S r0=1 ∑K r2=1 r2ξ (r1,r2). 5. Expected number of customers enter into the waiting hall E[CEWH] = ∑S r1=0 ∑K−1 r2=0 (K− r2)θr1ξ (r1,r2). 6. Expected waiting time of a customer in the waiting hall E[WT] = E[CWH] E[CEWH] 7. Probability that the server is busy P(busy) = ∑S r0=1 ∑K r2=1 ξ(r1,r2). 8. Probability that the server is idle P(idle) = 1 − P(busy) 9. The total expected cost value of the proposed model is defined as TCV = caE[psl]+ cbE[reorder] + ccE[perishable] + cdE[WT] where ca− Holding cost per unit, cb− Setup cost per unit, cc− Perishable cost per unit and cd− Waiting cost per customer. 14 Analysis of Finite Population Stochastic modeling Table 1: TCV for the case of SDAP and QDSP s 7 8 9 10 11 12 13 S 58 2.805974 2.805623 2.806103 2.807504 2.809919 2.813445 2.818190 59 2.806534 2.805602 2.805445 2.806146 2.807788 2.810462 2.814264 60 2.807988 2.806522 2.805778 2.805835 2.806769 2.808661 2.811598 61 2.810287 2.808327 2.807044 2.806509 2.806792 2.807966 2.810111 62 2.813383 2.810970 2.809190 2.808110 2.807794 2.808309 2.809726 63 2.817234 2.814403 2.812165 2.810584 2.809717 2.809625 2.810374 64 2.821800 2.818583 2.815924 2.813880 2.812506 2.811856 2.811991 6 Simulation Analysis In this section, the optimum cost analysis, monotonic behavior of some system char- acteristics are to be discussed by the numerical illustrations. This will be helpful to deliver a effective decision making polices for every inventory business tycoons. For knowing such curious results of our proposed model, we need to fix the value of the parameters and the cost values such that θ = 5, θ0 = 2, µ = 9, α = 0.9, γ = 0.07, β1 = 0.5, β2 = 0.5, S = 61, s = 10, N = 10, ca = 0.05, cb = 0.9, cc = 0.1, and cd = 7. Example 6.1. Optimum cost analysis This example briefly investigate the minimum optimal TCV for the category of both arrival and service processes of homogeneous and non-homogeneous cases as shown in Table (1)-(2). In Table (1), S ∈ B6458 and s ∈ B137 are used to find the minimal optimum TCV under the case of discussion between SDAP and QDSP. In this case, the TCV ∗ = 2.805445 and corresponding optimum S∗ = 59 and s∗ = 9 are obtained. Next, the output values of the case non-SDAP and non-QDSP are given in Table (2). Here, S ∈ B6357 and s ∈ B1610 are varied to get an optimum TCV. In this case, the TCV ∗ = 8.966159 and corresponding optimum S∗ = 60 and s∗ = 13 are obtained. As we expected due to the assumption of the proposed model, the case non-SDAP and non- QDSP have a higher TCV ∗ than the SDAP and QDSP case. That is, the minimal optimum TCV obtained in the case of SDAP and QDSP. Hence the arrival and service rates influence the cost value become a minimum one. k1 = 0, k2 = 0 k1 = 0, k2 = 0.6 Example 6.2. The variation of TCV under the parameter variation In this example, we describe the path of TCV with each parameter considered in the model. In such a way, the major objective of this example is discussed with the 15 K. Lakshmanan, S Padmasekaran, M Bhuvaneshwari, R Asokan Table 2: TCV for the case of non-SDAP and non-QDSP s 10 11 12 13 14 15 16 S 57 8.986034 8.976046 8.970983 8.970903 8.975952 8.986375 9.002534 58 8.985814 8.975121 8.969166 8.967979 8.971672 8.980449 8.994615 59 8.986685 8.975367 8.968613 8.966432 8.968907 8.976202 8.988575 60 8.988575 8.976700 8.969232 8.966159 8.967537 8.973497 8.984257 61 8.991414 8.979048 8.970942 8.967068 8.967457 8.972214 8.981518 62 8.995143 8.982342 8.973667 8.969073 8.968572 8.972242 8.980232 63 8.999705 8.986521 8.977338 8.972097 8.970793 8.973483 8.980284 scaling factors β1 and β2, because they are deciding factors whether the arrival and service processes are non-SDAP and non-QDSP or not respectively. In Table (3), the scaling factor β1 increases the total cost if it is increasing. That is, β1 increases means, the arriving customers in the system is increased. Subsequently, the sales of number of product in the inventory is raised. So the management is often ready to store or making reorder for their requirement. These jobs cause the increase of total cost. The same characteristics are holds the parameter θ. Simultaneously, when we are focusing the another scaling factor β2, more interestingly it reduces the TCV. If β2 increases means that the service time of an individual become reduced. So the number of customers leaves the system after a successful service completion of them is increasing. This helps to reduce the mean service time of a customer. So this is the reason for TCV is reduced if β2 increases. If β2 and µ are directly proportional to each other µ holds the same behavior as β2. Then the perishable parameter α1 affects the item life time. If α1 raises, the number of current stock level starts falling down. If it happens, the management has to store more number of products which cause the extra expenditure to maintain the system. So this expenditure cause the increase of total cost. Finally, the reorder intensity rate α minimize the total cost when it is increasing. The successive mean reorder time reduced means the number of available product of the system become positive. Therefore, the service completion will be done as soon as possible. Hence, all the parameters involved in Table (3) and Table (4) are satisfies their own properties. Example 6.3. Graphical Analysis • The scaling factors β1 and β2 shows the increasing/decreasing path due to its SDAP and non-SDAP, QDSP and non-QDSP. We observe that 0.2 ≤ β1 ≤ 1 the β2 curves deviation is high and 0.5 ≤ β2 ≤ 1.0 the β2 curves deviation is low for all β1 ∈ (0.2, 1). • The graph of expected waiting time is shown in Figure (2) when β1 and β2 are varying together. Here, the deviation of β2 curves coincides with the characteristics as we said in Figure (1). 16 Analysis of Finite Population Stochastic modeling Table 3: The variation of TCV under the parameter variation β2 θ α1 β1 0 0.5 1 µ 7 9 11 7 9 11 7 9 11 α 0 4.5 0.05 0.70 11.08 8.92 7.55 44.64 35.40 29.67 136.60 104.43 85.33 0.90 10.83 8.66 7.28 44.78 35.11 29.13 145.45 109.22 87.88 1.10 10.71 8.52 7.14 45.25 35.24 29.04 154.70 114.88 91.46 0.07 0.70 11.18 9.02 7.64 44.31 35.24 29.60 132.94 102.29 83.98 0.90 10.92 8.74 7.36 44.40 34.91 29.03 141.43 106.85 86.36 1.10 10.79 8.60 7.21 44.86 35.02 28.92 150.45 112.34 89.81 0.09 0.70 11.29 9.12 7.73 44.01 35.10 29.54 129.67 100.36 82.73 0.90 11.01 8.82 7.43 44.06 34.73 28.93 137.82 104.70 84.96 1.10 10.87 8.68 7.28 44.50 34.82 28.80 146.61 110.02 88.30 5 0.05 0.70 11.21 9.06 7.69 44.81 35.59 29.87 137.00 104.88 85.82 0.90 10.95 8.79 7.41 44.93 35.28 29.31 145.74 109.58 88.28 1.10 10.83 8.65 7.27 45.38 35.38 29.20 154.89 115.16 91.78 0.07 0.70 11.31 9.15 7.78 44.48 35.43 29.80 133.34 102.75 84.46 0.90 11.04 8.87 7.49 44.55 35.08 29.20 141.73 107.21 86.76 1.10 10.91 8.73 7.34 44.98 35.16 29.08 150.65 112.62 90.14 0.09 0.70 11.42 9.25 7.86 44.19 35.29 29.74 130.08 100.82 83.22 0.90 11.13 8.95 7.57 44.21 34.90 29.11 138.12 105.06 85.36 1.10 10.99 8.80 7.41 44.63 34.96 28.96 146.81 110.30 88.62 5.5 0.05 0.70 11.31 9.17 7.80 44.95 35.74 30.03 137.33 105.26 86.22 0.90 11.06 8.89 7.52 45.04 35.41 29.45 145.97 109.87 88.61 1.10 10.93 8.76 7.38 45.48 35.50 29.33 155.04 115.37 92.04 0.07 0.70 11.42 9.26 7.89 44.62 35.59 29.96 133.68 103.13 84.87 0.90 11.14 8.98 7.60 44.67 35.21 29.35 141.96 107.50 87.09 1.10 11.01 8.83 7.45 45.08 35.28 29.21 150.80 112.84 90.40 0.09 0.70 11.52 9.36 7.97 44.33 35.45 29.91 130.41 101.20 83.63 0.90 11.23 9.06 7.67 44.33 35.03 29.26 138.36 105.35 85.69 1.10 11.09 8.91 7.52 44.73 35.08 29.10 146.97 110.52 88.89 0.5 4.5 0.05 0.70 2.02 1.83 1.70 5.58 4.87 4.41 18.73 16.13 14.48 0.90 2.04 1.85 1.73 5.31 4.61 4.16 17.71 15.08 13.43 1.10 2.06 1.88 1.76 5.16 4.48 4.03 17.18 14.51 12.85 0.07 0.70 2.06 1.87 1.74 5.640 4.92 4.46 18.78 16.19 14.54 0.90 2.08 1.89 1.76 5.36 4.66 4.21 17.75 15.13 13.48 1.10 2.10 1.92 1.80 5.22 4.53 4.08 17.21 14.55 12.89 0.09 0.70 2.10 1.90 1.77 5.69 4.98 4.50 18.83 16.25 14.59 0.90 2.12 1.93 1.80 5.42 4.71 4.25 17.79 15.17 13.52 1.10 2.15 1.96 1.84 5.27 4.58 4.12 17.23 14.59 12.93 5 0.05 0.70 2.02 1.83 1.70 5.60 4.89 4.44 18.80 16.21 14.56 0.90 2.04 1.85 1.73 5.33 4.63 4.18 17.773 15.15 13.50 1.10 2.06 1.88 1.76 5.18 4.50 4.05 17.22 14.57 12.91 0.07 0.70 2.06 1.86 1.73 5.66 4.95 4.48 18.85 16.27 14.62 0.90 2.08 1.89 1.76 5.38 4.68 4.23 17.80 15.19 13.55 1.10 2.11 1.92 1.80 5.24 4.55 4.10 17.25 14.61 12.95 17 K. Lakshmanan, S Padmasekaran, M Bhuvaneshwari, R Asokan Table 4: The variation of TCV under the parameter variation β2 θ α1 β1 0 0.5 1 µ 7 9 11 7 9 11 7 9 11 α 0.09 0.70 2.106 1.904 1.769 5.719 5.003 4.532 18.908 16.327 14.678 0.90 2.125 1.931 1.801 5.441 4.737 4.276 17.845 15.242 13.596 1.10 2.154 1.967 1.843 5.294 4.600 4.148 17.283 14.649 12.999 5.5 0.05 0.70 2.029 1.832 1.699 5.618 4.918 4.460 18.860 16.278 14.637 0.90 2.046 1.855 1.727 5.347 4.656 4.205 17.818 15.205 13.562 1.10 2.071 1.888 1.765 5.202 4.519 4.076 17.266 14.616 12.965 0.07 0.70 2.068 1.868 1.733 5.678 4.970 4.506 18.913 16.335 14.693 0.90 2.087 1.893 1.763 5.403 4.706 4.250 17.854 15.250 13.609 1.10 2.114 1.928 1.803 5.256 4.568 4.121 17.293 14.655 13.008 0.09 0.70 2.107 1.903 1.766 5.737 5.022 4.552 18.966 16.392 14.749 0.90 2.127 1.931 1.798 5.457 4.755 4.294 17.891 15.296 13.656 1.10 2.156 1.967 1.840 5.309 4.617 4.165 17.321 14.695 13.051 1.0 4.5 0.05 0.70 1.021 0.981 0.957 1.290 1.190 1.124 3.460 3.200 3.026 0.90 1.124 1.086 1.064 1.329 1.228 1.160 3.231 2.976 2.805 1.10 1.207 1.173 1.153 1.382 1.282 1.215 3.129 2.882 2.716 0.07 0.70 1.045 1.004 0.979 1.313 1.210 1.142 3.489 3.224 3.047 0.90 1.152 1.113 1.090 1.355 1.251 1.181 3.261 3.001 2.827 1.10 1.239 1.203 1.183 1.410 1.307 1.239 3.160 2.908 2.739 0.09 0.70 1.069 1.027 1.002 1.335 1.230 1.161 3.518 3.248 3.068 0.90 1.179 1.140 1.116 1.380 1.274 1.202 3.290 3.026 2.849 1.10 1.270 1.233 1.212 1.438 1.333 1.262 3.191 2.935 2.763 5 0.05 0.70 0.998 0.955 0.929 1.283 1.182 1.114 3.469 3.209 3.036 0.90 1.101 1.060 1.035 1.322 1.219 1.150 3.239 2.985 2.814 1.10 1.185 1.147 1.125 1.374 1.272 1.204 3.136 2.890 2.725 0.07 0.70 1.022 0.978 0.951 1.306 1.202 1.132 3.498 3.234 3.057 0.90 1.129 1.086 1.061 1.347 1.241 1.170 3.269 3.010 2.836 1.10 1.216 1.177 1.154 1.402 1.297 1.227 3.167 2.916 2.748 0.09 0.70 1.045 1.000 0.973 1.328 1.221 1.150 3.527 3.258 3.078 0.90 1.156 1.113 1.087 1.372 1.264 1.191 3.298 3.035 2.858 1.10 1.247 1.207 1.183 1.429 1.322 1.250 3.198 2.942 2.771 5.5 0.05 0.70 0.978 0.932 0.904 1.277 1.175 1.106 3.477 3.218 3.045 0.90 1.081 1.037 1.010 1.315 1.211 1.141 3.246 2.992 2.822 1.10 1.166 1.125 1.100 1.367 1.264 1.194 3.143 2.897 2.732 0.07 0.70 1.001 0.955 0.926 1.300 1.194 1.124 3.506 3.242 3.066 0.90 1.108 1.063 1.036 1.340 1.233 1.161 3.276 3.017 2.844 1.10 1.197 1.154 1.129 1.395 1.289 1.217 3.174 2.923 2.755 0.09 0.70 1.024 0.977 0.947 1.322 1.214 1.142 3.534 3.266 3.086 0.90 1.135 1.089 1.060 1.365 1.256 1.181 3.305 3.042 2.866 1.10 1.227 1.183 1.157 1.423 1.314 1.240 3.204 2.949 2.778 18 Analysis of Finite Population Stochastic modeling Figure 1: Impact of TCV on β1 vs β2 • Figure (3) explores expected present stock level of the system on the combination of β1 vs β2. Both β1 vs β2 reduce the E[psl] when they are increasing. Here, the deviation of β2 curves is high when 0.2 ≤ β1 ≤ 1. • The parameters α and α1 are affects the E[WT] as shown in Figure 4. In this graph, the beta curve has the high deviation with themselves and low deviation with α1. • The E[WT] is shown in Figure 5 if θ and µ are increasing together. The θ curves are decreasing when µ is increasing and it means that the increased service rate cause less mean service time of an individual.Therefore, the E[WT ] is decreased. If θ and µ are inversely proportional each other, θ reacts against µ. • The average waiting time of a customer is discussed for the case of θ vs K in Figure 6 and µ vs K in Figure 7. As we have enough discussion about θ and µ on the E[WT], we shall move to analyses the impact of K.When the number of finite source population is increases, for the E[WT], the θ curve will be straight line. Example 6.4. Impact of E[psl], E[reorder] and E[perishable] with the parameter variation This example describes the important system performance measures, E[psl], E[reorder] and E[perishable] are to be discussed with the parameter analysis of α, α1, θ, µ and β2 as shown in Table 5-7. As per the scaling factor, β2 = 0, β2 = 0.5 and β2 = 1 are to be explored in Table 5, 6 and 7 respectively. If we increase the reorder rate, the expected present stock level increases. For every replenishment, there are Q items replaced as it reaches the system. So it makes the E[psl] is increasing when it is increase. When α 19 K. Lakshmanan, S Padmasekaran, M Bhuvaneshwari, R Asokan Figure 2: Impact of E[WT] on β1 vs β2 Figure 3: Impact of E[psl] onβ1 vs β2 20 Analysis of Finite Population Stochastic modeling Figure 4: Impact of E[WT] on α vs α1 Figure 5: Impact of E[WT] on µ vs θ 21 K. Lakshmanan, S Padmasekaran, M Bhuvaneshwari, R Asokan Figure 6: Impact of Impact of E[WT] on K vs θ Figure 7: Impact of E[WT] on µ vsK 22 Analysis of Finite Population Stochastic modeling is increasing, for the value of µ = 7 E[reorder] behaves first increasing and then de- creasing but at µ = 11 it is increasing only. Further, for both µ, the E[perishable] will increase. Here, the perishable quality of the products depends on the number of present stock level of the system. The parameter α1 affects the E[psl] to fall down if it is increasing. Perishable prod- ucts starts deterioration process depending the existing current stock level. So it is decreased. Since the items in the inventory storage system are perished, the system requires more number of products to provides the sales service. Hence the expected reorder level is increased. Here the raise of a perishable rate obviously influence the increase of E[perishable] . Then the parameter θ changes the E[psl] and E[reorder] by direct variation where as with E[perishable] it varies by indirect variation. For every arrival, there will be an unit in the system getting down when they leave the system. To fulfill such required number of items, there must be a reorder needed. Since the in- ventory reduces by the more sales, there must be less number of items remaining in the inventory storage place. This cause the E[perishable] become less. More interestingly, as we predicted earlier, the intensity rate µ is inversely propor- tional to each E[psl], E[perishable]. If we increase µ, each of them starts falling down. If mean service time of individual customer too short, number of inventory falls down fast and less inventory requires more reorder and less number of perishable items. 23 K. Lakshmanan, S Padmasekaran, M Bhuvaneshwari, R Asokan Table 5: Impact of E[psl], E[reorder and E[perishable] with the parameter variation β2 θ α1 E[stock] E[reorder] E[perishable] µ 7 11 7 11 7 11 α 0 4.5 0.05 0.70 24.9645 22.6523 0.3136 0.4386 1.2482 1.1326 0.90 26.3157 24.3173 0.3138 0.4543 1.3158 1.2159 1.10 27.2531 25.5090 0.3089 0.4602 1.3627 1.2755 0.07 0.70 24.4147 22.2220 0.3277 0.4503 1.7090 1.5555 0.90 25.8047 23.9103 0.3293 0.4679 1.8063 1.6737 1.10 26.7722 25.1230 0.3252 0.4751 1.8741 1.7586 0.09 0.70 23.9075 21.8168 0.3408 0.4614 2.1517 1.9635 0.90 25.3337 23.5268 0.3439 0.4809 2.2800 2.1174 1.10 26.3299 24.7591 0.3408 0.4893 2.3697 2.2283 5 0.05 0.70 24.9645 22.6523 0.3188 0.4503 1.2482 1.1326 0.90 26.3157 24.3173 0.3191 0.4665 1.3158 1.2159 1.10 27.2531 25.5090 0.3142 0.4726 1.3627 1.2755 0.07 0.70 24.4147 22.2220 0.3330 0.4621 1.7090 1.5555 0.90 25.8047 23.9103 0.3347 0.4803 1.8063 1.6737 1.10 26.7722 25.1230 0.3307 0.4877 1.8741 1.7586 0.09 0.70 23.9075 21.8168 0.3463 0.4733 2.1517 1.9635 0.90 25.3337 23.5268 0.3495 0.4934 2.2800 2.1174 1.10 26.3299 24.7591 0.3464 0.5022 2.3697 2.2283 5.5 0.05 0.70 24.9645 22.6523 0.3211 0.4553 1.2482 1.1326 0.90 26.3157 24.3173 0.3214 0.4718 1.3158 1.2159 1.10 27.2531 25.5090 0.3165 0.4780 1.3627 1.2755 0.07 0.70 24.4147 22.2220 0.3353 0.4672 1.7090 1.5555 0.90 25.8047 23.9103 0.3371 0.4856 1.8063 1.6737 1.10 26.7722 25.1230 0.3330 0.4932 1.8741 1.7586 0.09 0.70 23.9075 21.8168 0.3486 0.4785 2.1517 1.9635 0.90 25.3337 23.5268 0.3519 0.4988 2.2800 2.1174 1.10 26.3299 24.7591 0.3488 0.5078 2.3697 2.2283 24 Analysis of Finite Population Stochastic modeling Table 6: Impact of E[psl], E[reorder and E[perishable] with the parameter variation β2 θ α1 E[stock] E[reorder] E[perishable] µ 7 11 7 11 7 11 α 0.5 4.5 0.05 0.70 18.5347 15.4582 0.2337 0.2938 0.9267 0.7729 0.90 20.5232 17.5264 0.2533 0.3280 1.0262 0.8763 1.10 22.0246 19.1546 0.2662 0.3532 1.1012 0.9577 0.07 0.70 18.2647 15.2755 0.2384 0.2973 1.2785 1.0693 0.90 20.2574 17.3398 0.2589 0.3323 1.4180 1.2138 1.10 21.7661 18.9683 0.2725 0.3582 1.5236 1.3278 0.09 0.70 18.0051 15.0982 0.2430 0.3007 1.6205 1.3588 0.90 20.0014 17.1583 0.2644 0.3366 1.8001 1.5442 1.10 21.5169 18.7869 0.2787 0.3631 1.9365 1.6908 5 0.05 0.70 18.2833 15.0507 0.2366 0.2977 0.9142 0.7525 0.90 20.2883 17.1217 0.2573 0.3339 1.0144 0.8561 1.10 21.8085 18.7631 0.2712 0.3609 1.0904 0.9382 0.07 0.70 18.0201 14.8771 0.2412 0.3009 1.2614 1.0414 0.90 20.0281 16.9432 0.2628 0.3380 1.4020 1.1860 1.10 21.5549 18.5840 0.2773 0.3657 1.5088 1.3009 0.09 0.70 17.7668 14.7084 0.2456 0.3041 1.5990 1.3238 0.90 19.7774 16.7695 0.2681 0.3420 1.7800 1.5093 1.10 21.3101 18.4095 0.2834 0.3705 1.9179 1.6569 5.5 0.05 0.70 18.1713 14.8689 0.2379 0.2993 0.9086 0.7434 0.90 20.1830 16.9399 0.2591 0.3364 1.0092 0.8470 1.10 21.7112 18.5861 0.2733 0.3642 1.0856 0.9293 0.07 0.70 17.9110 14.6992 0.2424 0.3025 1.2538 1.0289 0.90 19.9254 16.7649 0.2644 0.3404 1.3948 1.1735 1.10 21.4596 18.4102 0.2794 0.3689 1.5022 1.2887 0.09 0.70 17.6605 14.5342 0.2468 0.3056 1.5894 1.3081 0.90 19.6770 16.5946 0.2697 0.3443 1.7709 1.4935 1.10 21.2168 18.2387 0.2854 0.3736 1.9095 1.6415 25 K. Lakshmanan, S Padmasekaran, M Bhuvaneshwari, R Asokan Table 7: Impact of E[psl], E[reorder and E[perishable] with the parameter variation β2 θ α1 E[stock] E[reorder] E[perishable] µ 7 11 7 11 7 11 α 1 4.5 0.05 0.70 12.2895 10.2694 0.1562 0.1954 0.6145 0.5135 0.90 14.2804 12.1335 0.1792 0.2281 0.7140 0.6067 1.10 15.9157 13.7102 0.1973 0.2549 0.7958 0.6855 0.07 0.70 12.1776 10.1919 0.1586 0.1974 0.8524 0.7134 0.90 14.1615 12.0488 0.1822 0.2306 0.9913 0.8434 1.10 15.7933 13.6209 0.2007 0.2578 1.1055 0.9535 0.09 0.70 12.0681 10.1157 0.1610 0.1994 1.0861 0.9104 0.90 14.0450 11.9654 0.1851 0.2330 1.2640 1.0769 1.10 15.6733 13.5331 0.2041 0.2606 1.4106 1.2180 5 0.05 0.70 11.2674 9.0422 0.1473 0.1796 0.5634 0.4521 0.90 13.2040 10.7934 0.1709 0.2123 0.6602 0.5397 1.10 14.8221 12.3059 0.1899 0.2400 0.7411 0.6153 0.07 0.70 11.1733 8.9823 0.1494 0.1812 0.7821 0.6288 0.90 13.1024 10.7265 0.1734 0.2144 0.9172 0.7509 1.10 14.7161 12.2343 0.1928 0.2424 1.0301 0.8564 0.09 0.70 11.0811 8.9232 0.1514 0.1828 0.9973 0.8031 0.90 13.0026 10.6606 0.1759 0.2164 1.1702 0.9594 1.10 14.6120 12.1636 0.1958 0.2448 1.3151 1.0947 5.5 0.05 0.70 10.7714 8.4426 0.1426 0.1709 0.5386 0.4221 0.90 12.6732 10.1258 0.1662 0.2033 0.6337 0.5063 1.10 14.2751 11.5941 0.1856 0.2311 0.7138 0.5797 0.07 0.70 10.6852 8.3903 0.1445 0.1723 0.7480 0.5873 0.90 12.5794 10.0669 0.1686 0.2051 0.8806 0.7047 1.10 14.1766 11.5305 0.1883 0.2332 0.9924 0.8071 0.09 0.70 10.6007 8.3388 0.1464 0.1738 0.9541 0.7505 0.90 12.4872 10.0088 0.1709 0.2069 1.1239 0.9008 1.10 14.0798 11.4676 0.1910 0.2354 1.2672 1.0321 26 Analysis of Finite Population Stochastic modeling 7 Conclusion The finite source population is considered to explore the non-SDAP and non-QDSP in the SQIS. The generalization of homogeneous and non-homogeneous arrival and service processes are given in the steady state of the model. Also, the comparative dis- cussion is made in the numerical investigations.The illustrations given in the examples enhance minimum optimal total cost for the QDSP category. The SDAP will increase the number of units arriving to the inventory system. This increased units of arrival will produce the more sales of the inventory. When we are focusing the development of the inventory business, the first step has to be initialized to attract the customers towards the system. For such process, displayed stock level will assure the increase of customers in the inventory system. And Maintaining the sufficient current stock level will play the crucial role for the development of an inventory system. Simultaneously, the QDSR contribute the reduce of waiting time of a customer in the system. 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