CHAPTER 1 INTRODUCTION. 1.1 Introduction. 1.2 Inventory System

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1 CHAPTER 1 INTRODUCTION 1.1 Introduction The study on multi-server inventory systems generally assumes the servers to be homogeneous so that the individual service rates are same for all the servers in the system. This assumption may be valid only when the service process is mechanically or electronically controlled. In an inventory system with human servers, the above assumption can hardly be realized. It is common to observe that servers render service to identical jobs at different service rates. This reality leads to modelling such multi-server inventory systems with heterogeneous servers, i.e., the service time distributions may be different for different servers. This is incorporated in this thesis. 1.2 Inventory System Inventory is the stock of any item or resource used in an organisation. An inventory system is the set of policies and controls that monitor levels of inventory and determine what levels should be maintained, when stock should be replenished and how large orders should be. The real need for analysis of an inventory system was first recognized in industries that had a combination of production scheduling problems and inventory problems. That is, situations where items were produced in lots and then stored at a factory warehouse. 1

2 1.2.1 Characteristics of Inventory System Every inventory system emptied by a demand process and refilled by a replenishment process, both of which significantly influence the system behaviour. Inventory systems can be classified as 1. Demand process: The number of units will need to be withdrawn from inventory for some use during a specific period. Deterministic: The rate of demand is known with certainty and to be constant overtime. Stochastic: The demand cannot be predicted, but the demand in any period is a random variable rather than a known constant. 2. Lead time: Lead time is the amount of time between the placement of an order to replenish inventory and the receipt of the goods into an inventory. Deterministic: Lead time is known with certainty. Stochastic: Lead time is a random variable. 3. Review policy: Review policy refers to whether the current inventory level is being monitored continuously or periodically. Continuous review: The inventory level is monitored continuously, and order is placed, as soon as the stock level falls down to the prescribed reorder point. e.g. super market Periodic review: The inventory level is checked at discrete intervals and ordering decisions are made only at those times even if the inventory level dips below the reorder point. e.g. regular stock-taking for a grocery store. 4. Excess demand: Backog: The excess demand is not lost, but instead, which is hold and it can be satisfied when the next normal delivery replenishes the inventory. Lost sales: The excess demand cannot be wait for the next normal delivery. 5. Inventory life(changing inventory): Limited life time of an item Obsolescene: It typically occurs when an item has been replaced by a better version. e.g. electronic components, maps cameras, etc., 2

3 Perishability: A perishable item is one that has constant utility up until an expiration date (which may be known or unknown), at which point the utility drops to zero. 6. Planning horizon: The planning horizon can be a single period, a finite number of periods, or can have infinite length. The increasing interest in practical inventory management (measured for example by the number of downloads of inventory-related items on the Internet) and in inventory research (see the growing number of participants at inventory conferences and sessions) clearly express the importance of a revival of the best traditions when some of the best minds of economics and management research focused at least part of their interest on inventories. The earliest derivation of Simple lot size formula of an inventory system was obtained by Harris [1915]. It is often referred to as Wilson s formula, since it was derived by R. H. Wilson as an integral part of an inventory control scheme. The first full length boom to deal with the inventory problems was that of Raymond [1931]. It explains how various extensions of the simple lot size model can be used in practice. Everyone agrees that the golden age of inventory research was in 1950 s. They were the first researchers to provide a rigorous analysis of a multi period stochastic inventory problem. Three significant books on the theory research: Whitin [1953], Arrow et al. [1958], Hadley and Whitin [1963], Axsäter [2006]. Poisson demands were considered by Scarf [1958], Karlin and Scarf [1958], Morse [1958] and Galliher et al. [1959]. A valuable review of the problem in probability theory of storage is given by Gani [1957]. Hadley and Whitin [1963] deals with the application of mathematical models to practical situations. The cost analysis of different inventory systems is given by Naddor [1966]. Tijms [1972] gives a detailed analysis of inventory systems under (s, S) policy. The most quoted book on managerial applications is Peterson and Silver [1979]. Schwarz [1973] and Zheng [1987] listed the known properties including zero inventory ordering property and last minute ordering property. For a clear and precise description of inventory management the reader is referred to Silver et al. [1998], Gilrich [1984] provided a survey on dynamic inventory problems and implementable problems. The excellent survey on perishable inventory system is given by Nahmias [1982]. Love [1979], Bartmann and Beckmann [1992] and Zipkin [2000] summarized several results in storage systems. 3

4 1.3 Ordering Policies There are two basic types of review ordering policies: Continuous review and Periodic review. These policies are defined based on the inventory position( = on-hand inventory + orders outstanding - back orders) of the item. Three most common ordering policies in connection with continuous review inventory systems are often denoted by (R, Q) policy, (s, S) policy and (S 1, S) policy. The periodic review inventory system has two common ordering policies, namely, (t, S) policy and (t, Q) policy. 1. (R, Q) Policy: (R, Q) is a fixed replenishment point/fixed replenishment quantity inventory policy. When the inventory level on-hand falls below a certain replenishment point, R, the inventory level will generate a replenishment order for a certain quantity, Q, of this product. When using this policy, the reorder point field is set as the trigger level. The reorder/order up to quantity field will be the exact number of units reordered. 2. (s, S) Policy: (s, S) is a minimum/maximum inventory policy. When the inventory level on-hand falls below a minimum, s, the inventory will generate a request for a replenishment order that will restore the on-hand inventory to a target, or maximum number, S. When using this policy, the reorder point field is the minimum, or trigger level. The reorder/order up to quantity field is the maximum, or the number to which the inventory level is restored. The main difference between (s, S) and (R, Q) is that the (s, S) takes into account exactly how far below the reorder level the inventory is when the request for replenishment is generated. 3. (S 1, S) Policy: This policy simply means that we order an item whenever a demand occur, which is known as order-up-to S or base stock policy 4. (t, S) Policy: We only can place an order at scheduling periods of length t, so as to bring back the inventory level up to S. 5. (t, Q) Policy: This policy place an order for Q items at epochs of t interval length. Arrow et al. [1951] introduced the (s, S) form of inventory policy. The policies are designed for retailers of finished goods, who face economies of scale when placing orders with their suppliers. Scarf [1960] showed that there exists an optimal (s, S) policy for a single item with stochastic demand. Veinott [1966] gives a detailed review of the work 4

5 carried out in (s, S) inventory systems up to Porteus [1971] analysed inventory systems with piecewise linear concave ordering costs. He showed that a generalized (s, S) policy is optimal for a multi-period periodic review inventory system under some mild assumption. Tijms [1972] gives a detailed analysis of inventory systems under (s, S) policy. A practical treatment of the (s, S) inventory systems can be found in the books by Silver and Peterson [1985] and Tijms [1986]. Algorithms for a continuous review (s, S) inventory system in which the demand is according to a versatile Markovian point process is given by Ramaswami [1981]. Kalpakam and Arivarignan [1984] discussed the (s, S) policy inventory models in which demands form a semi-markov process. An inventory system with two ordering levels and random lead times is analysed by Thangaraj and Ramanarayanan [1983]. Kalpakam and Arivarignan [1990] introduced multiple reorder level policy. Kalpakam and Sapna [1993] analysed an (s, S) ordering policy in which items are procured on an emergency basis during stock out period. Kalpakam and Sapna [1994] discussed a continuous review (s, S) inventory policy with stochastic lead time and provided a detailed cost analysis. For a comprehensive review of continuous review (s, S) and (R, nq) policies can be found Noblesse et al. [2014]. 1.4 Queueing System Queueing theory is a collection of mathematical model of various queueing systems. It is very useful in many practical applications in areas such as, e.g. telephone exchange, traffic control, manufacture systems, inventory systems and communication systems, supper market, at a petrol station, at computer systems, etc., Characteristics of Queueing System 1. Arrival process: Customers arrival for a service. Input source: The size represents the total number of potential customers who will require service. According to source According to numbers According to time Pattern of arrivals at the system: It is classified into two categories. Static arrival process: The control depends on the nature of arrival rate (random or constant). 5

6 Dynamic arrival process: It is controlled by both service facility and customers Behavior of arrivals: Patient: If a customer, on arriving at the service system stays in the system until served, no matter how much he has to wait for service. Impatient: The customer waits for a certain time in the queue and leaves the service system without getting service due to certain reasons. Balking: The customer decides not enter the huge length queue. Reneging: The customer may enter the queue but after he loses patient then he decides to leave. Jockying: Customer may move from one queue to another queue. 2. Service mechanism: The service mechanism consists of one or more service facilities, each of which contains one or more parallel service channels, called servers. The time elapsed from the commencement of service to its completion for a customer at a service facility is referred to as the service time. Single server-single queue. Single server-several queues. Several servers-single queue. Several servers-several queues. Service facilities in a series. 3. System capacity: There can be a limitation with respect to the number of customers in the system. The queueing system may be of finite or infinite capacity. 4. Queue discipline: The queue discipline refers to the order in which customers of the queue are selected for service. The first in first out (FIFO) is the most commonly used service discipline following the natural law. last in first out (LIFO), is also followed in many occasions. Such a queue discipline is followed in inventory systems, where there is no obsolescence of stored units, since the last arrived unit is easier to reach. In a service in random order (SIRO), the customers are selected at random for service. 6

7 In the priority (PRI) queues, the customers are assigned priorities while they enter the system. Pre-emptive priority: The higher priority customers served first, the interruption allowed, and then the service of lower priority customers started at that point where the service was stop. Non pre-emptive priority: the higher priority customer does not interrupt the service of lower priority customers and have to wait till the service of the lower priority customer has been completed Kendell s Notation Any queueing system is represented by the notation introduced by Kendall [1951] as follows: A/S/m/B//k/SD. A : Arrival processes S : Service time distribution m : Number of servers B : System capacity k : Population size SD : Service discipline Johannsen s Waiting times and number of calls (an article published in 1907 and reprinted in post office Electrical Engineers Journal, London, October, 1910) introduced the first paper on queueing theory. But the method used in this paper was not mathematically exact. Therefore, from the point of view of exact treatment, the paper that has historic importance is Erlang [1909]. A Danish mathematician who studied telephone traffic congestion problems in the first decade of the 20th century. In this paper, he laid the foundation for the place of Poisson distribution in queueing theory. Until 1940, the majority of the contribution to queueing theory was made by people active in the field of telephone traffic problems. After the second world war, the field of operations research rapidly developed and queueing applications were also found in production planning, inventory control and maintenance problems. In this period, much theoretically oriented research on queueing problems were done. In the fifties and sixties, the queueing theory reached a very high mathematical level. A probabilistic approach to the analysis was initiated by Kendall [1951, 1953]. 7

8 For a complete bibliography of research on this period, see Syski [1960], Saaty [1961, 1966], Bhat [1969]. Wolff [1982] proved and popularized the Poisson arrival see time average (PASTA) principle. Starting with the introduction of phase-type probability distributions, Neuts [1975] developed an analysis techniques that extends and modifies the earlier transform method to multi variables and makes it amenable for an algorithmic solution. For matrix analytic method in queueing theory, see Neuts [1978, 1989], Sengupta [1989], Ramaswami [1990, 2001] and He [2014]. The use of phase type distributions in the representation of system elements and the matrix analytic method in their analysis has significantly expanded the scope of queueing systems for which usable results can be derived. For bibliographies on vacation models topic, see Doshi [1986], Takagi [1991], Alfa [2003] and Tian and Zhang [2006]. More recent ones appearing in journals provide bibliographies for further study on retrial queues, Yang and Templeton [1987], Falin [1990], Kulkarni and Liang [1997], Falin and Templeton [1997] and Artalejo and Gomez-Corral [2008]. The interested reader may see Newell [1971], Cooper [1981], Takagi [1991], Latouche and Taylor [2002], Breuer and Baum [2005], Gross et al. [2008] and Alfa [2010]. 1.5 Inventory System with Service Facility Inventory maintained without service facility, available at the demand epoch, the customer need not have to wait. That time, a queue can be formed only when the inventory level becomes zero and lead time is positive. But an inventory maintained at service facilities, the service time is not negligible. In this situation, a queue will be formed even when the item is available and the inventory is depleted by one at the moment of service completion which service rate is usually higher than the demand rate. The real life applications are supply bicycles which need assembly of its parts, supply food items that need heating or garnishing, computer organizations that need installation and basic services, etc., Berman et al. [1993] introduced the concept of an inventory management system at a service facility which uses one item of inventory for each service provided. They assumed that both demand and service rates are deterministic and constant as such queues can form only during the stock-outs. They determined optimal order quantity that minimizes the total expected cost rate. Although the paper of Sigman and Simchi- Levi [1992] published earlier than the paper of Berman et al. [1993], the formers cited the work of later and hence we give the credit to the later. Sigman and Simchi-Levi 8

9 [1992] studied a single server inventory system in which the demands arrive according to Poisson process, exponentially distributed replenishment time and arbitrary distribution for service times. Using light traffic heuristic method, they derived closed-form solution of the model. Berman and Kim [1999] discussed inventory policies for inventory management at service facilities. They analysed a problem in a stochastic environment where customers arrive according to a Poisson process. The service times are exponentially distributed with mean inter arrival time assumed to be larger than the mean service time. They derived the optimal policy under the condition that the order quantity is known. Berman and Sapna [2000] dealt with inventory management at service facilities. They assumed Poisson arrivals of demands, arbitrarily distributed service times and zero lead time. They analysed the system with the restriction that the waiting space is finite. They derived the optimal ordering quantity that minimizes the long-run expected cost rate under a specified cost structure. Berman and Sapna [2001] addressed the problem of optimally controlling service rates for an inventory system of service facility with positive lead time and Poisson demands for given maximum inventory and reorder levels. Also they discussed an inventory system where speeding up or slowing down the service rate is possible. Elango [2001] presented a Markovian inventory system with instantaneous supply of orders at a service facility. The service time assumed to have exponential distribution with parameter depending on the number of waiting customers. Arivarignan et al. [2002] considered a perishable inventory system with service facility with the arrival of customers forming a Poisson process. Each customer requires single item, which is delivered through service of random duration having exponential distribution. Arivarignan and Sivakumar [2003] considered an inventory system with arbitrary distribution for interoccurrence of demands, exponential service time and exponential lead time. The impact of negative customers on the inventory control problem at service facility considered by Sivakumar and Arivarignan [2005] and Paul et al. [2007]. Krishnamoorthy and Islam [2004] analysed an (s, S) inventory system with postponed demands. They assumed that the arrival of customers to form a Poisson process and lead time is exponentially distributed. Krishnamoorthy and his co-authors (Krishnamoorthy and Jose [2005], Krishnamoorthy et al. [2006c], Krishnamoorthy et al. [2006b], Narayanan et al. [2008], Krishnamoorthy and Narayanan [2011],Krishnamoorthy et al. [2010], Krishnamoorthy and Narayanan [2013]) investigated the behaviour of service systems with an attached 9

10 inventory. A survey on inventory system with positive service time can be found in Krishnamoorthy et al. [2011]. Paul et al. [2006, 2008] analysed a continuous review (s, S) inventory system at service facility, where in an item is demanded by a customer which is issued after performing service on the item. Sivakumar and Arivarignan [2008] emphasized an elaborate work on stochastic inventory system with service facilities. A sequence of paper dealt by Yadavalli and his co-authors (Yadavalli et al. [2007], Yadavalli et al. [2008], Yadavalli et al. [2011], Yadavalli et al. [2012]). Nair and Jacob [2014] analysed an (s, S) inventory system with positive service time and retrial demands by considering the multi-server inventory system. They assumed that the demands arrive according to a Poisson process and service time distribution is exponential. Recent papers in an inventory system with service facility are Shophia Lawrence et al. [2013], Hamadi et al. [2014], Jenifer and Sivakumar [2014] and Amirthakodi and Sivakumar [2014]. 1.6 Inventory System with Vacations In several situations, the server is unavailable to the customers due to server s failure, may be engaged in other works such as maintenance or serving secondary customers, or may just go away and may not be waiting. The aim of studying the queueing model with vacation is, by utilizing the idle time of the server, by which the total average cost involved may be minimized. Applications arise naturally in call centres with multitask employees, customized manufacturing, telecommunication and computer networks, maintenance activities, production and quality control problems, etc.,. Some important types of vacation models are discussed below under vacation termination and vacation start-up. 1. Multiple Vacations: When the system becomes empty, server starts a vacation and the server keeps on taking vacations until, on return from a vacation, at least on customer is present. 2. Single Vacation: The server taking vacation exactly one vacation at the end of each busy period. 3. Exhastive Service Discipline: Each time the server becomes available, he works in a continuous manner until the system becomes empty. 10

11 4. Single Service Discipline: Systems with a vacation period beginning after each service completion. 5. Non-Exhaustive Service Discipline: The server may take a vacation even when some customers are presented in the system. Bernoulli: It is a generalization of both exhaustive and single service disciplines. That is, after each service completion, the server takes a vacation with probability p and starts a new service with probability 1 p. If the system is empty, after a service completion or vacation completion, server always takes a vacation and after any vacation if customers are present server resumes service. Gated: In a gated system, as soon as the server returns from a vacation it places a gate behind the last waiting customer. It then begins to serve only the customers who are within the gate, based on some rules of how many or how long it could serve. Limited: A fixed limit of K is place on the maximum number of customers that can be served before the server goes on vacation. Remaining customer will wait in the queue. The server leaves for vacation either: When the system is empty. when K customers have been served. 6. Synchronous Vacation(Multi-server System): All server may take a vacation together(single or multiple) after serving exhaustively. 7. Asynchronous Vacation(Multi-server System): Some or partial servers may take (single or multiple) vacation individually and independently, if those servers are idle. Analysis of queueing systems with vacations to the server is motivated by the study of cyclic queues and Miller [1964] was the first to study such a system. Miller analysed a system in which the server goes for a vacation (a rest period ) of random length whenever it becomes idle. He also considered a system in which the server behaves normally but the first customer arriving to an empty system has a special service time. Comprehensive surveys on vacations can be found in Doshi [1986, 1990]. In this collection, see also books by Takagi [1991], Medhi [2003] and the recent book by Tian and Zhang [2006]. One of the fundamental objective of vacation models is to investigate the optimal control 11

12 of a system in which a cost structure is assumed, see the recent surveys of Tadj and Choudhury [2005]. Many researchers have studied queueing system with different kinds of vacations. In that particularly, Multiple vacation models are used to approximate delays in computer systems, where maintenance segments are run when no jobs are in queue, and in manufacturing applications, where preventive maintenance is performed when there are no jobs in queue. The interested readers may see Vinod [1986], Lucantoni et al. [1990],Kao and Narayanan [1991], Gray et al. [1997], Tyagi et al. [2002], Krishnakumar and Pavai Madheswari [2005], Yue et al. [2011], El-Taha [2011],Dimitriou [2012] and Jeyakumar and Senthilnathan [2012]. An inventory system with server vacation has received very little attention in the literature. In the case of random lead times, the concept of vacations to the server during dry period is introduced in inventory system by Danial and Ramanarayanan [1987]. Danial and Ramanarayanan [1988] studied an (s, S) inventory system in which the server takes a rest when the inventory level is zero. They assumed that the interarrival times between successive demands, the lead times, and the rest times are assumed to follow arbitrary distributions. Narayanan et al. [2008] considered an inventory system with random positive service time. The server took multiple vacations whenever there was no customer waiting in the system or the inventory level was zero. Customers arrived at the service station according to a Markovian arrival process and the service time for each customer had a phase-type distribution. They assumed correlated lead time for the orders and an infinite waiting hall for the customers. The customers who wait for service may renege after a random time. Under the above assumptions, they analysed the level dependent quasi birth-death process. Krishnamoorthy and Narayanan [2011] considered a production inventory system with server vacation. They assumed that the production process was according to Markovian production process and that service times for each customer had a phase-type distribution. Sivakumar [2011] considered an inventory system with retrial demands and multiple server vacation. He assumed independent exponential distributions for inter-demand times, lead times, inter-retrial times and server vacation times. He also assumed that all these events are mutually independent. He adopted a multiple vacation policy. Jayaraman et al. [2012] considered a perishable inventory system with postponed demands in which the server takes multiple vacations. They assumed that demand time points form 12

13 a Poisson process. The life time of each item, vacation time of the server and the lead time follow independent exponential distribution. Padmavathi et al. [2014] considered a continuous review stochastic (s, S) inventory system with Poisson demands and exponentially distributed lead time. They gave a comparative study of single and modified vacation policies. 1.7 Inventory System with Retrial Queues Retrial queues (or repeated calls, queues with returning customers) are very pervasive in real-life systems. Retrial queueing systems are characterized by the feature that arriving customers who find the server busy, join the retrial group (orbit) and attempts service again at random intervals and continues to do so until the server is found idle. These queueing models have been used to model many problems in telephone switching systems, telecommunication networks, computer networks, etc.,. There are two types of retrial policy considered in the literature of queueing systems: 1. Classical Retrial Policy: The intervals between successive repeated attempts are exponentially distributed with rate nα, when the orbit size is n. (The probability of a repeated attempt depends on the number of orbiting customers) 2. Constant Retrial Policy: The retrial rate is independent of the number of customers (if any) in the orbit, i.e., the retrial rate is (1 δ 0n )α, where δ ij denotes Kroneker s delta. Most retrial queues assume to involve a classical retrial policy. Cohen [1957] considered a markovian multiple-server queueing system where the customers may renege, balk or conduct retrials. Aleksandrov [1974] considered an M/G/1/1 system with retrials and the retrial times follow exponential distribution. He analysed a queue-length process in such a system in the steady state. Waiting times and other aspects of queueing systems with retrials considered by Falin [1979]. Hanschke [1987] expressed the stationary distribution in terms of hyper geometric function for the case of two servers with repeated attempts. The constant retrial policy was introduce by Foyalle [1986], who modelled a telephone exchange system. Farahmand [1990] calls this discipline a retrial queue with FCFS orbit. Choi et al. [1993] generalized the constant retrial policy by considering an M/M/1 retrial queue with general retrial times where the customer only at the head of the orbit may 13

14 attempt retrials from the orbit. Artalejo et al. [2001] discussed the M/M/c retrial queue with constant retrial rate, where the system was analysed a quasi-birth and death process. For detailed overviews of the main results and methods, the reader is referred to the survey papers by Yang and Templeton [1987], Falin [1990] and the books by Falin and Templeton [1997], Artalejo and Gomez-Corral [2008]. For most recent references see the bibliographical overviews in Artalejo [1999a], Artalejo [1999b], Artalejo [2010] and Zhang and Wang [2013]. Moreover, Artalejo et al. [2006] introduced the concept of retrial demands in inventory. They assumed positive lead time and unsatisfied customers joined to the orbit and retry for their demand under constant retrial policy. Ushakumari [2006] considered an (s, S) inventory system with random lead time and repeated demands of unsatisfied customers form the orbit at a constant rate. Krishnamoorthy and Jose [2007] analysed three different retrial inventory systems with positive service time and positive lead time. The authors assumed Poisson arrivals, exponentially distributed lead time and exponentially distributed inter-retrial time. Sivakumar [2008] considered a two-commodity substitutable retrial inventory system with joint ordering policy. Paul et al. [2008] considered a perishable inventory system with service facilities. They assumed that demand points from a Markovian arrival process, service time follows a phase type distribution and the life time for the items, lead time and inter-retrial times follow exponential distribution. Sivakumar [2009] considered a perishable inventory system with retrial demands and the demands are generated from a finite-source. He assumed exponential life times for the stored items, exponential lead time and exponential retrial times. Lopez-Herrero [2010] provided an analysis for performance measures related to waiting time and first passage time distributions of the retrial inventory models having finite retrial group. Sivakumar [2011] considered single server retrial inventory system with multiple vacation, in which arrival follows Poisson and lead time, inter-retrial time and vacation time have independent exponential distribution. Yadavalli et al. [2011] investigated a multiserver retrial inventory system of perishable item in the stock with service facility in which apart from the usual demands. They assumed another stream of demands called negative demands, who removes one ordinary customer from the tail of the queue. Yadavalli et al. [2012] treated finite source retrial inventory system with multi-server at service facility. 14

15 1.8 Heterogeneous Servers The study on multi server inventory systems generally assume the servers to be homogeneous in which the individual service rates are the same for all the servers in the system. Multi servers of different service rates have many practical aspects in modelling real systems, but many literature on this class is limited to the servers having identical service rates as it simplifies the analysis. Models for many practical scenarios, for example, nodes in wireless systems serving different mobile users, nodes in the telecommunications network with links of different capacities, servers formed with different processors as a consequence of system updates, communications network supporting communication channels of different transmission rates, multiprogramming computer system which spools its output for printing on a set of printers of different speeds, or scheduling jobs on functionally equivalent processors of a local computer network, etc., involve heterogeneous servers. Those servers have heterogeneous service rates because one cannot expect to work at a constant rate. Queueing modellers considered this aspect and study queueing systems with heterogeneous servers following pioneering work of Morse [1958] who considered the situation of some hyper-exponential distributions for service time with parallel service channels. Saaty [1961] expressed the concept of heterogeneous servers in finding the time dependent solution of multi server Markovian queue. Krishnamoorthy [1963] considered a Poisson queue with two-heterogeneous servers with modified queue disciplines. The steady-state solution, transient solution and busy period distribution for the first discipline and steady-state solution for the second discipline ate obtained. Singh [1970] extended the work of Krishnamoorthy [1963] on two-server Markovian queueing system with balking and compared it with a corresponding homogeneous system. Seth [1977] considered, where a customer reneges if his wait in the queue exceeds T, a random variables. Neuts and Takahashi [1981] observed that for queueing systems with more than twoheterogeneous servers analytical results are intractable and only algorithmic approach could be used to study the steady state behaviour of the system. Larsen and Agrawala [1983] analysed a scheduling discipline designed for a heterogeneous multiple servers system. Walrand [1984] considered the following slow server problem. The queueing system has two exponential servers with different rate and has a single queue with Poisson arrivals. The objective is to minimize the expected queue length. They proved 15

16 that the optimal policy is a threshold type, i.e., the fast server should be always used and the slow server should be used if the queue length exceeds a certain value. Hajek [1984] discussed parallel queueing systems with heterogeneous exponential servers under some conditions of holding cost, he showed that an optimal control policy has a switch curve structure. Chaudhry et al. [1987] considered a two-heterogeneous server Markovian system such that both server will serve customers in batches of size. Koyanagi and Kawai [1995] considered on optimization problem for a parallel queueing system with two-heterogeneous servers. Madan et al. [2003] studied a two server queue with Bernoulli schedules and a single vacation policy in which the two servers provide heterogeneous exponential service to customers. They obtained the steady state probability generating functions of the system size for various states of the servers. Krishnakumar and Pavai Madheswari [2005] analysed M/M/2 queueing system heterogeneous servers, where the servers go on vacation in the absence of customers waiting for service. They studied the stationary queue length distribution along with their means via the rate matrix. Yue et al. [2009] further considered the model studied by Krishnakumar and Pavai Madheswari [2005]. They obtained the explicit expression of the rate matrix and proved the conditional stochastic decomposition results of the stationary queue length and waiting time. Yue and Yue [2010] discussed an M/M/2 queueing system with two heterogeneous servers under variant vacation policy, where the two servers may take together at most J-vacations when the system is empty. Kumar [2010] considered a catastrophic-cumrestorative M/M/2 queueing model with heterogeneous servers. El-sherbiny [2012] incorporated balking and general balking function in a Markovian queueing system with reneging and two heterogeneous servers and derived steady-state solution using iteration method. Krishnamoorthy and Sreenivasan [2012] displayed a queueing model with twoheterogeneous servers, where one server remains idle but the other goes on vacation in the absence of waiting customers. He and Xiuli-Chao [2014] studied a tandem queueing system with k servers and no waiting space in between. 1.9 Outline of the Thesis This thesis is divided into ten chapters. Chapter 1 gives an introduction to the literature and ideas surrounding the inventory and queueing system. Chapter 2 contains concepts of Stochastic processes, Discrete and Continuous time Markov Chains, Phase- 16

17 type distribution, Markov arrival processes and Quasi birth-death processes. Chapter 3, we consider a (s, S) inventory system with multiple vacations and service facility for two heterogeneous servers. The customers arrive according to a Poisson process. The demanded items are delivered to the demanding customers after performing some service which is distributed as an exponential distribution for each server. The lead times for the orders are assumed to have independent and identical exponential distributions. Both servers avail vacation whenever the inventory level reaches zero or the customer level reaches zero or both. At the end of a vacation period, the service commences if there is a positive inventory and at least one customer is in the system; Otherwise, the server takes another vacation immediately and continues in the same manner until he finds both inventory level and the customer level are positive (multiple vacations). This process holds good for both servers. The vacation times of both servers are assumed to be independent and identically distributed exponential random variables. The joint probability distribution of the number of customers in the system, inventory level and server status is obtained in the steady state. The steady state analysis of the inventory model is performed using matrix analytic methods. Some important performance measures are obtained and the optimality of an expected total cost rate is shown through numerical illustrations. Chapter 4, we have incorporated all the assumptions that are made in chapter 3. Further we assumed that the customers arrive according to a Markovian arrival process and two parallel servers who provide heterogeneous phase type services to customers. We calculate the expected total cost rate using the system performance measures. The joint probability distribution of the number of customers in the system, inventory level and server status is obtained in the steady state. Some important performance measures are obtained and investigate the effect of variations of costs numerically. Chapter 5 deals a continuous review inventory system with a finite number of homogeneous sources of customers and multiple vacations of two heterogeneous servers. We have assumed that two heterogeneous servers who provide phase type services to customers. The inventory is replenished according to an (s, S) policy and the lead time follows an exponential distribution. Both servers can avail vacation whenever the inventory level reaches zero or the customer level reaches zero or both. At the end of a vacation period, the service commences if there is a positive inventory and at least one customer in the system; Otherwise, the server takes another vacation immediately and continues in the same manner until he finds both inventory level and the customer level 17

18 are positive(multiple vacations). This process holds good for both servers. The vacation times of both servers are assumed to be independent and identically distributed exponential random variables. The joint probability distribution of inventory level,number of customers in the system and server status is obtained in the steady state. Some important system performance measures are derived and the long-run total expected cost rate is also calculated. The results are illustrated with some numerical examples. Chapter 6, we consider a retrial inventory system with two heterogeneous servers and multiple vacations for servers. We assume the customers arrive according to a Markovian arrival process. We use (s, S) ordering policy to replenish the stock. The lead time is assumed to be exponential. If the inventory level drops to one, one server goes for vacation and when the inventory level drops to zero, another server goes for vacation. The vacation time for the two servers are exponentially distributed with different rates. The customer demands unit item at a time and the demanded items are delivered after a random service. Any arriving customer, who finds a server is idle, joins the service. The service time for the two servers are exponentially distributed with different rates. If both servers are busy and or both servers are in vacation, then the arriving customer joins the orbit of infinite size. These orbiting customers retry after an exponential time for their service. An arriving customer gets service from server - 1 whenever both servers are in idle. Using matrix geometric method, we obtain the joint probability distribution of the number of customers in the orbit, server status and inventory level in the steady state. Some important performance measures are obtained and the optimality of an expected total cost rate is shown through numerical illustration. Chapter 7, we consider a finite retrial (s, S) inventory system with multiple vacations for two heterogeneous servers. The customers arrive according to a Markovian arrival process. The demanded items are delivered to the demanding customers after performing some service which is distributed as phase-type distribution for each server. The lead time for the order is assumed to have an exponential distribution. The vacation times of both servers are assumed to be independent and identically distributed exponential random variables. The arriving customer who finds both servers are busy or both servers are on vacation, joins an orbit of finite size. These orbiting customers retry for their demand after a random time, which is assumed to be exponential distribution. An arriving customer gets service from server - 1 whenever both servers are in idle. The joint probability distribution of the inventory level, the number of customers in the orbit and the server status is obtained in the steady state. Various system performance 18

19 measures are derived and the results are illustrated numerically. Chapter 8 deals a continuous review (s, S) inventory system with two heterogeneous servers, say, server - 1, server - 2. Server - 1 serve for primary customers and server - 2 is served for both primary and feedback customers. The primary customer s arrival according to a Markovian arrival process (MAP) and service time for both servers has exponential distribution. The lead time is assumed to be exponential distribution. The primary customers, who finds either two servers are busy or no item is in the stock, waits in the finite waiting hall. If the waiting hall is full, then arriving, customer considered to be lost. After the completion of service the primary customer will decide either to join the orbit, which is an infinite size, for additional service or leaves the system according to a Bernoulli trail. These orbiting (feedback) customers compete for their service according to constant retrial policy, and the service times for these feedback customers are assumed to be exponential distribution. After completing the service for feedback customer s server - 2 becomes idle or busy for primary customer. If the server - 2 completes the service for a primary customer, then he searches the customers in the orbit or idle according to a Bernoulli trail. The results are illustrated numerically. In the final chapter we present the summary of the thesis. 19