Introduction 2.1 Queues with server s vacations

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1 Introduction A concise literature review on queues with server s vacations, working vacations, different control policies, batch arrival queues, queues with server breakdowns, twophase queueing systems with second phase of optional service, two essential phases of service, two-phase systems with server breakdowns, retrial queueing systems and customers impatience are presented in this chapter in a chronological order till to date. 2.1 Queues with server s vacations Queues with vacations or simply called vacation models attracted great attention of queueing researchers and became an active research area. Many studies on vacation models were published from the 1970's to the mid 1980 ' and were summarized in two survey papers by Doshi and Teghem, respectively, in Stochastic decomposition theorems were established as the core of vacation queueing theory. In the early 1990 Takagi published a set of three volume books entitled Queueing Analysis. One of Takagi's books was devoted to vacation models of both continuous and discrete time types and focus mainly on M/G/1 type and Geo/G/l type queues with vacations. Takagi's book certainly advanced further research and wide applications of vacation models Many authors have studied the utilization of server s idle time in queueing systems. These queueing systems have got wide applications in computer, communication, production and other stochastic systems. Miller (1964) was the first to study an M/G/1 queueing system where the server is unavailable during some random length of time (referred to as vacation). Levy and Yechiali (1975) have found server s idle time utilization in the M/G/1 queue based on the assumption that as the queue becomes empty, the server takes vacation of exponential distribution during which he does some secondary work. They have given the expressions for the Laplace-Stieltjes transform of the system measures, like occupation period, vacation period, and waiting time. 18

2 Levy and Yechiali (1976) have obtained expressions for the expected system length in an M/M/S queueing model with vacation and have shown that the mean system size is a linear function of the mean vacation time. Michel Scholl and Klein rock (1983) studied the M/G/1 queue with behaviour of the delay and rest periods under FCFS order of service. They obtained the second moment of the waiting time in this system with rest periods and random order of service. They obtained the Laplace-Stieltjes transform of the distribution function of the waiting time in the above system with rest periods and non-preemptive LCFS order of service. Doshi (1985) studied M/G/1 model with variable vacations. He assumed that the probability of the vacation lengths depends on the number of consecutive vacations taken during a single busy cycle and the server must continue to take vacations until he returns to find customers waiting. Fuhrmann and Robert Cooper (1985) studied stochastic decomposition in the M/G/1 queue with generalized vacations. They considered a class of M/G/1 queueing models with a server who is unavailable for occasional intervals of time. Servi (1986) studied a D/G/1 queue with vacations. He gave an example of this model where many data switching systems have processors with arrival streams of regularly (deterministically) spaced tasks requiring attention (e.g. directing the bytes to the appropriate outgoing line). He defined and analyzed a model for investigating the waiting time at the primary queue in such systems in terms of the primary task arrival rate, the probability distribution of the vacation duration and the processor s service schedule. Doshi (1986) presented a survey on queueing systems with vacations. In this survey, he gave an overview of some general decomposition result. He has also shown how other related models can be solved in terms of the results for the basic models. 19

3 Jacob and Madhusoodanan (1988) examined the transient behaviour of the infinite capacity M/G/1 model with batch arrivals and server vacations. Using renewal theoretic arguments, they obtained explicit expressions for the system size probabilities at arbitrary epochs and also obtained expressions for the probability distribution of the virtual waiting time in the queue at any time t. Kella and Yechiali (1988) studied an M/G/1 queue with single and multiple server vacations under both the preemptive and non-preemptive regimes. Ramaswamy & Servi (1988) studied various aspects of the server busy period of an M/G/1 vacation model with a Bernoulli schedule. Simple expressions are derived for (i) the conditional joint distribution of the busy period and number in the system at the end of a busy period given the number in the system at the beginning of the busy period (ii) the ergodic distribution of the number in the system at a busy period initiation epoch and (iii) the distribution of the depletion time at service initiation epochs. They also found expressions in the real domain for the key ingredients of the busy periods. Tian et. al. (1989) derived probability distribution of the queue length at arrival epochs in the GI/M/1 queue with exhaustive service and multiple vacations using matrixgeometric method. Further, they explained the limiting behaviour of the continuous time queue length processes and obtained the probability distributions for the waiting time and busy periods. Leung and Eisenberg (1990) analyzed an M/G/1 queue with multiple server vacations and gated time-limited service. Kella (1990) considered the M/G/1 queue with server s vacation where they assumed that the decision of going for vacation depends on the number of vacations already taken through a random outcome and obtained Laplace- Stieltjes transforms of the results under the expected long-run average cost criterion with linear holding costs, fixed setup cost and a concave piecewise linear reward function for being on vacation. 20

4 Alfa & Gupta (1991) studied an M/G/1 queue model for queue length in a manufacturing process by taking the server vacations in to consideration. They also showed that given appropriate information on costs of waiting time, processing time and an opportunity cost of failure, an optimal inspection policy can be obtained. Takagi (1992) studied an M/G/1/N queue with the server s vacations. These results have been applied to a polling model with a finite population of customers, which can be used for the performance evaluation of token-ring networks connecting several computers, each of which supports a finite number of iterative users. Choi and Han (1994) studied a G/M (a/b) /1 queue with multiple vacation discipline. They obtained explicit expressions for queue length probabilities at arrival time points and arbitrary time points simultaneously by using supplementary variable method. Katayama (1995) analyzed a cyclic-service tandem queue with multiple server vacations and exhaustive services. He derived explicit formulae for the mean total sojourn time for an N ( 2) stage tandem queue with vacations and mean waiting time in the first stage. The results are useful to the performance analysis for message processing in packet switching systems. Katsaros and Langaris (1995) presented an analysis of a structured priority queue with an arbitrary number of customer classes and multiple server vacations. They obtained the Laplace transforms of the joint distributions of the system states and the elapsed or remaining (or vacation) time for both transient and steady states. Also, they obtained a formula for the amount of unfinished work. Choudhury (1996) has found the steady state solution of an M/M/1 queue with a random setup time and indicated some actual situations in queueing and inventory analysis, where such models could be useful. 21

5 Feinburg and Kim (1996) observed the bi-criterion optimization of an M/G/1 queue with a server that can be switched on and off. The first criterion is the average number of customers in the system and the second is the average operating cost per unit time. Borthakur and Choudhury (1997) studied the steady state behavior of M X /M/1 queue and generalized vacations. They obtained the queue size distributions at stationary point of time, departure point of time and vacation initiation point of time. Also, they showed that the departure point of time queue size distribution decomposes into three random variables one of which is the queue size of the standard M X /M/1 queue. Eitan Altman and Arie Hordijk (2000) considered the optimal open-loop control of vacations in queueing systems. The controller has to take actions without state information. They first considered the case of a single queue, in which the question is when vacations should be taken so as to minimize, in some general sense, workloads and waiting times. Then they considered the case of several queues, in which service of one queue constitutes a vacation for others. Choudhury and Borthakur (2000) discussed and derived analytically some stochastic decomposition results for a class of batch arrival Poisson queues with a grand vacation process at various points of time. Also, they derived new results as generalizations of known results for the vacation model. Madan and Saleh (2001) studied in three papers a single server queue with exponential service and deterministic vacations, deterministic service with exponential vacations and deterministic service with deterministic vacations, assuming Bernoulli vacation schedules. Zhang and Tian (2001) observed a Geo/G/1 queueing system with multiple adaptive vacations. Tian and Zhang (2002) studied a discrete time GI/Geo/1 queue with server vacations. Using matrix-geometric method they obtained the explicit expressions for the stationary 22

6 distributions of queue length and waiting time, and also demonstrated the stochastic decomposition property. Tian and Zhang (2003) analyzed a GI/M/C type queueing system with vacations in which all servers take vacations together when the system becomes empty. These servers keep taking synchronous vacations until they find customers waiting in the system at a vacation completion instant. The vacation time is a phase-type distributed random variable. They obtained explicit expressions for the stationary probability distributions of the queue length at arrivals and the waiting time. Tsuyoshi Katayama and Kaori Kobayashi (2003) considered an M/G/l-type, two-phase queueing system, in which the two phases are attended alternatively and exhaustively by a moving single-server according to a batch service in the first phase and an individual service in the second phase. They showed that the two-phase queueing system reduces to a new type of single-vacation model with non-exhaustive service and derived the joint distribution of the queue length in each phase and the remaining service time. Alfa (2003) analyzed a batch of single server vacation queues, which have single arrivals and non-batch service in discrete time. He has shown that use of discrete time approach to the study of some vacation models is more appropriate and makes the models much more algorithmically tractable. These results are applicable to all the classes of vacation queues, provided, the inter-arrival, service, vacation and operational time can be represented by a finite Markov chain. Madan et al. (2003) analyzed a two server queue with Bernoulli schedules and a single vacation policy. They considered two models; in one model after completion of a service both servers can take a vacation while in the other only one may take a vacation. They obtained the steady state probability generating functions of system size for various states of the servers. Jau-Chuan Ke (2003) studied a single removable server in a G/M/1 queueing system with finite capacity where the server applies an N policy and takes multiple vacations 23

7 when the system is empty. They developed the steady-state probability distributions of the number of customers in the system and illustrated analytically. Madan and Abu Al-Rub (2004) analyzed an M/D/1 queue with optional server vacations based on exhaustive service. Unlike other vacation policies, they assumed that only at the completion of service of the last customer in the system, the server has the option to take a vacation or to remain idle in the system waiting for the next customer to arrive. They derived the explicit steady state results for the probability generating functions of the queue length, the expected numbers of the customers in the queue and the expected waiting time of the customers. Madan and Choudhury (2004) analyzed an M x /G/1 queue with a Bernoulli vacation schedule under admissibility policy. They obtained the steady-state queue size distribution at random point of time as well as at a departure epoch and also derived some important performance measures of this system. Tian and Zhang (2006) studied a queueing system with C servers and a threshold type vacation policy. In their policy, when a certain number d < C servers become idle at a service completion instant, these d servers take a synchronous vacation of random length together. After each vacation, the number of customers in the system is checked. If that number is N or more, these d servers will resume serving the queue; otherwise, they will take another vacation together. They called it as (d, N)-Threshold Policy. Chae et al. (2006) observed the GI/M/1 queueing system with single exponential vacation. They derived the probability generating function of the stationary queue length, and Laplace Stieltjes transform of the stationary FIFO sojourn time and proved the stochastic decomposition property. Jau-Chuan Ke(2007) observed the operating characteristics of an M x /G/1 queueing system with N-policy and at most J vacations. They developed the system size distribution at a random epoch as well as at a departure point of time and derived the 24

8 busy period distribution, the idle period distribution, and the queue waiting time distribution. Sokol Ndreca, Benedetto Scoppola (2007) studied a discrete time single server system with generic distribution of the number of arrivals in a time slot, geometric distribution of the service time and two classes of customers. They gave a complete description of these systems and are able to compute the expected waiting time of the customers of the two classes, and solving a boundary value problem. Wang et al. (2007) have discussed the Maximum entropy analysis of the M x /M/1 queueing system with multiple vacations and server breakdowns. By a comparison between the exact expression for the expected delay time and an approximate expected delay time based on the maximum entropy estimate, they argued that their maximum entropy estimate is sufficiently accurate for practical purposes. Banik and Gupta (2008) studied a finite-buffer single server queue with single (multiple) vacation(s) under batch arrival Markovian process. The service discipline is E-limited variation where the server serves until either the system is emptied or randomly chosen limits of L customers have been served. They obtained queue length distributions at service completion and vacation termination epochs, arbitrary epochs, and pre-arrival epochs and discussed performance measures. Chae et al. (2008) studied a discrete time GI/Geo/1 queueing system in which the server takes exactly one geometric vacation each time the system empties. They have presented the probability generating functions of the stationary queue length and the stationary FCFS sojourn time. Also they studied the connection between the results of their study and other previous established results. Iyer and Sikdar (2008) derived the equilibrium distribution at pre-arrival and arbitrary epochs and the waiting time distribution in a GI/M/1 vacation queueing system with dependence between the service time of each customer and the subsequent inter-arrival times. 25

9 Ma et al. (2008) observed an M/G/1 queue with the multiple adaptive vacations and pure decrementing service policy based on classical M/G/1 queueing model. They have derived the probability generating function of stationary queue length using an embedded Markov chain method and also obtained the Laplace-Steiltjes transform of stationary waiting time and presented special cases of the model and shown similarity with existing results. Vasanta Kumar.V, K. Chandan.K et. al (2010) Studied Two-phase M/Ek/1 queueing system with N-policy for exhaustive batch service with gating, and server start-ups and breakdowns. Explicit expressions for the steady state distribution of the number of customers in the system are obtained and also derived the expected system length. The total expected cost function is developed to determine the optimal threshold of N at a minimum cost. Sharma D.C. (2012) took a machine repairable system with spares and two repairmen where the partial server vacation is applied. In their system, the first repairman never takes vacations and always available for serving the failed units. The second repairman goes to vacation of random length when number of failed units is less than N. At the end of vacation period, this repairman returns back if there are N or more failed units/machine accumulated in the system. Otherwise this repairman goes for another vacation. Steady state probabilities, reliability measures are given. 2.2 Queues with N-policy Yadin and Naor (1963) were first to study the concept of N-policy. They studied an M/G/1 queueing system and obtained the optimal value of the queue size at which to start on a single server, assuming that the form of the policy is to turn on the server when the queue size reaches a certain number, N and to turn him off when the system size is empty. Heyman (1968) analyzed the economic behaviour of the system M/G/1 when costs are associated with the length of the queue, the operation of the server, and the change of 26

10 the service state. The model is somewhat artificial in the sense that startup costs are incurred in the absence of any distinct startup times. Generally, the addition of startup times makes the M/G/1 queue difficult to analyze. Baker (1973) studied the M/M/1 queue with exponential startups. He derived the optimum number of customers present that minimizes the mean time cost when startup times are zero and non-zero respectively. Borthakur et al. (1997) extended Baker s model with exponential startup time to general startup time. Taghem (1987) observed the N-policy single removable server queueing system with finite capacity for the M/G/1 queue. Kella (1989) presented detailed discussions concerning N-policy queueing systems with vacations. Takagi (1990) studied the time-dependent joint process of the server state, the queue size and the elapsed time in that state for exhaustive service M/G/1 system with multiple vacations, single vacations, exceptional first service times and a combination of N-policy and setup times. Medhi and Templeton (1992) analyzed the steady state behaviour of an M/G/1 queue under N-policy and with a general startup time. Lee and Park (1992) studied an N- policy M/G/1 system with early setup: the server starts with setup when m < N customers are waiting. If there are still less than N customers are waiting after setup, he waits until N customers accumulate in the system. Takagi (1993) presented a steady state analysis of M/G/1/K queues which combined N-policy and setup times before service periods. The queue length distributions and the mean waiting times are obtained for an exhaustive service system, the gated service system, the E-limited service system and the G-limited service system and also provided numerical examples. 27

11 Wang and Huang (1997) derived the closed form solutions for the M/E k /1 queueing system with N-policy and reliable server. In another paper, Wang and Huang (1995 a) dealt with the economic behaviour of a removable server in the N-policy M/E k /1 queueing system with finite capacity. They derived the expressions for the probability mass functions of the number of customers in the system.also, they constructed the cost function for the total expected cost per unit time, and determined the optimal operating policy at a minimum cost. Lee and Park (1997) studied a queueing system in which the server is deactivated as soon as the system empties. The server is reactivated and starts a setup when m customers accumulate in the queue. After the setup, if there are less than N ( m) customers in the queue, the server remains dormant in the system until the number of customers reaches N. If N or more customers are in the queue after the setup, the server begins to serve the customers immediately. They called this policy (m, N)-policy. They derived queue length distribution, mean queue length and proposed a linear cost model. Choudhury (1998) analyzed the queue size distribution of an M/M/1 queue under N- policy with general setup time. Choudhury (1998) studied a general class of control operating policy with a random setup time, where with T-policy and N-policies are considered. He derived the queue size distribution at different points of time of the M/G/1 queue and demonstrated the decomposition property. Gupta (1999) studied an N-policy queueing system with startup time, finite source and warm spares and derived closed form expressions for recursively calculating the stationary distribution and presented an efficient algorithm to find the stationary probability distribution of the number of customers in the system. Wang and Ke (2000) provided a recursive method, using the supplementary variable technique, to obtain the exact steady-state solutions for the N-policy M/G/1 queueing system with finite and infinite capacity. 28

12 Ke (2001) presented the optimal control of an M X /G/1 queue with two types of generally distributed random vacation: type 1 (long) and type 2 (short) vacations. He has derived important system performance measures. He constructed the total expected cost function to determine the joint optimal threshold (Q*, N*) that minimizes the cost. Lee and Ahn (2002) analyzed the MAP/G/1 queue under N-policy with a single vacation and setup. They derived the generating function of the queue length at an arbitrary time and at departure time. Eugene Ke (2003) observed a single removable server with finite capacity along with N-policy and takes multiple vacations when the system is empty in a G/M/1 queueing system. He obtained the distributions of the number of customers in the queue at pre-arrival epochs and at arbitrary epochs, as well as the distributions of the waiting and the busy period. Tadj and Ke (2003) presented the optimal control of a bulk service queueing system under N-policy. If the number of customers in the system at a service completion is larger than some integer r, then the server starts processing a group of customers. If on the other hand it is smaller than r, then the server goes through an idle period and waits for the line to grow up to some integer N (N r). They derived the system characteristics as well as constructed the total expected cost function and determined the optimal threshold r and N that yields minimum cost. Zhang and Tian (2004) studied N threshold policy for the GI/M/1 queue method, and obtained the stationary distributions of queue length and waiting time. Tadj et. al. (2006) studied an optimal management policy for a bulk service queueing system with random setup time operating under N-policy with server vacations based on Bernoulli schedules. Moreno (2008) studied a discrete-time single server queueing system whose arrival stream is a Bernoulli process and service times are generally distributed that generalizes the concept of the N-policy, in the sense of turning the server on at the beginning of 29

13 each cycle with probability θ i, whenever i ( 1) customers are present in the queue. He derived the probability distributions of the lengths of the different periods in a cycle and the distributions of the time spent waiting in the queue as well as system. Liu et al. (2010) studied M/G/1 queueing system under N-policy with general startup/closedown times. They derived the recursion expressions of system size distribution for two different N-policy M/G/1 queueing systems with general startup/closedown times. Purohit G.N, Madhu Jain et.al (2012) investigated an M/M/1 retrial queue with constant retrial rate, unreliable server and threshold based recovery with state dependent arrival rates. They measured various system characteristics. The effect of various parameters on the system performances measures such as expected number of customers in the system and mean waiting time in the system is studied. 2.3 Batch arrival queueing systems Baba (1986) analyzed an M X /G/1 queueing system with multiple vacations. He derived the general queue length distribution at an arbitrary time and also obtained the waiting time and busy period distributions under multiple vacation policy using supplementary variable technique. Lee and Srinivasan (1989) studied M X /G/1 queueing system under two different control policies. For both the policies, they derived the mean waiting time of an arbitrary unit for a given value of m and found the stationary optimal m-policy which minimizes the expected cost per unit time in the long run, under this cost structure. Rosenberg and Yechiali (1993) observed the M X /G/1 queue without server vacations and with multiple vacations of the server under the LIFO service discipline. They derived the explicit formulae for the Laplace-Stieltjes transform, mean total expected cost. 30

14 Lee et al. (1995) have extended the stochastic decomposition result to a more general class of N-policy M [X] /G/1 queue with single and multiple vacations and also introduced the notion of grand vacation process. They have proved that the probability generating function of the queue size can be decomposed into three independent random variables, one of which is the queue size of the standard M X /G/1. Borthakur and Choudhury (1997) and Choudhury (1997) have studied the concept of vacation period along with batch arrival for Poisson queues. Choudhury (1998) observed a batch arrival Poisson queue with a random setup time, where the concept of random batch arrival, a setup time and a vacation period are introduced which is an extension to M/M/1 queue with a random setup time studied by Choudhury (1997). Choudhury (2000) analyzed an M X /G/1 queueing system with a vacation period and which consists of an idle period and a random setup period. He derived explicit expressions for the system state probabilities and some performance measures of the queueing system. Choudhury (2002) observed an M X /G/1 queueing system with a vacation time under single vacation policy, where the server takes exactly one vacation between two successive busy periods. He derived the steady state queue size distribution at different points of times as well as the steady state distributions of busy period and unfinished work. Ke (2003) presented the optimal control of M [X] /G/1 queueing system with two types of generally distributed random vacations and startup times. He presented the total expected cost function per unit time to determine the suitable thresholds of Q and N at a minimum cost. 31

15 Hur and Ahn (2005) analyzed a batch arrival queue with three types of idle periods: threshold, multiple vacations and single vacations and with setup time. Madan and Al-Rawwash (2005) studied a single server queue with batch arrivals and general service time distribution with optional server vacation. They obtained the steady state results for the number of customers in the queue, the average number of customers and the average waiting time in the queue beside other studies. Arumuganathan and Malliga (2005) observed a M X /G (a,b) /1 queueing system with multiple vacations and repair of service station on request by a leaving batch of customers and setup time. They presented expected system length, the busy period, the idle period and a cost model with numerical results. Tadj et. al. (2006) presented an optimal management policy for a bulk arriving queueing system with random setup time under Bernoulli vacation schedule and N-policy. They studied the discrete time parameter and continuous time parameter stochastic processes and designed a linear cost structure. Choudhury (2008) derived the queue size distribution of an M X /G/1 queue with a random setup time and with a Bernoulli vacation under a restricted admissibility policy. Anantha Lakshmi et al. (2008) studied the N-policy M X /M/1 queueing system with a removable, non-reliable server and startup times. They obtained various system performance measures and established the stochastic decomposition property. Also, they presented the optimal control policy under a liner cost structure. Ke and Lin (2008) analyzed the M X /G/1 queueing system in which the server operates N-policy and a single vacation with server breakdowns. The maximum entropy approach is used to examine the steady state probability distribution because the exact probability distribution of the variant vacation system is difficult to be obtained. 32

16 2.4 Queueing systems with two essential phases of service Two-phase queueing system with two essential phases of service was first introduced by Krishna and Lee (1990). They considered the exhaustive service with and without gating for the M/M/1 queueing system and derived the sojourn time distribution and its mean for an arbitrary customer. Doshi (1991) studied an M/G/1 queueing system in which each customer receives two services and derived the transforms of the queue length at departure epochs, queue length in steady state and sojourn time of an arbitrary customer. He considered both gated and exhaustive services. Selvam and Sivasankaran (1994) first introduced two-phase queueing system with server vacations and analyzed the model using the Laplace-Stieltjes transforms. Kim and Chae (1998) analyzed a single server two-phase queueing system with N- policy. First phase of service is batch service and the second phase of service is individual. They obtained the system size distribution and showed that the system size decomposes into three random variables. Further, they derived the system sojourn time and mean waiting time by heuristic inspection. They considered a linear cost structure and derived the optional stationary operating policy. Madan (2001) considered Bernoulli schedule server vacations for a single server queue in which the server provides a two-stage heterogeneous service with different general service time distributions to the incoming customers. He obtained the steady state probability generating function of the queue length for various states of the server and obtained results for some particular cases. Also, he cited some important applications of the day to day life situations. Kim and Park (2003) analyzed single server two-phase queueing system with N-policy. They obtained the system size distribution and proved that the system size decomposes into three random variables and provided system sojourn time. Also, they gave the 33

17 analysis of gated service and derived the condition under which the optimal operating policy is achieved. Choudhury and Madan (2005) analyzed a two-stage batch arrival queueing system with a modified Bernoulli vacation schedule under N-policy. They derived the queue size distribution at a random epoch as well as at a departure epoch under the steady state conditions and demonstrated the existence of stochastic decomposition property to show that the departure point queue size distribution of this model can be decomposed into the distributions of three independent random variables. Dimitriou and Langaris (2008) studied a two-phase model where all arriving customers are queued up in a single ordinary queue. After completion of the first phase of service the customer either proceeds to the second phase or joins the retrial box from where he retires, after a random amount of time, to find the server available, and to complete his second phase of service. The same authors (2010) generalized this model by allowing the server breakdowns and repairs in both phases of service, and incorporating the server startup in order to start serving a retrial customer in the second phase of service. 2.5 Two-phase queueing systems with server breakdowns Choudhury and Deka (2008) studied the steady-state behaviour of an M/G/1 retrial queue with an additional second phase of optional service subject to breakdowns occurring randomly at any instant while serving the customers. This model generalizes both the classical M/G/1 retrial queue subject to random breakdowns as well as M/G/1 queue with second optional service and server breakdowns. They have derived some important performance measures of this model. Anantha Lakshmi.S, Afthab Begum M.I et.al(2008) presented optimal strategy analysis of N-Policy queueing system with a removable and non-reliable server. various system measure and stochastic decomposition property are obtained using generating functions. Sensitivity analysis is also presented. 34

18 Choudhury and Tadj (2009) observed the steady-state behaviour of an M/G/1 queue with an additional second phase of optional service subject to breakdowns occurring randomly at any instant while serving the customers, and delayed repair. They have made extensive analysis of the joint distributions of the state of the server and number of units present in the system, i.e., queue size, stationary queue size distribution at a departure epoch, besides busy period distribution and waiting time distribution and also calculated some reliability indices from the joint distributions. Choudhury et al. (2009) analyzed the steady-state behaviour of an M X /G/1 with an additional second phase of optional service and unreliable server, which consists of a breakdown period and a delay period under N-policy. They derived the queue size distribution at random epoch and departure epoch as well as various system performance measures. They derived a simple procedure to obtain optimal stationary policy under a suitable linear cost structure. Choudhury et al. (2011) studied an M/G/1 queueing system with two phases of service and Bernoulli vacation schedule for an unreliable server, which consists of a breakdown period and a delay period, under N-policy and a random setup time. They derived the queue size distribution at different points of time, delay busy period distribution and optimal N-policy. Vasanta Kumar et al. (2011 a, 2011b) studied the optimal control policy of two-phase, N-policy M X /M/1 and M X /E k /1 queueing systems with server startup and breakdowns, respectively. They formulated the total expected cost function and obtained the optimal threshold of N at a minimum cost. Sensitivity analysis is carried out through numerical illustrations. Yue, D et al. (2011) analyzed a two-phase queueing system with impatient customers and multiple vacations. The customers may get impatience not only when the server is on vacation but also is busy carrying out the first essential service. They obtained the closed form expressions foe various performance measures including the mean system 35

19 sizes for various states of the server, the average rate of balking, the average rate of repairing and the average rate of loss. 2.6 Balking Haight (1957) was the first to study the notion of customer impatience appeared in the queuing theory. He considered a model of balking for M/M/1 queue in which there is a greatest queue length at which an arrival would not balk. This length was a random variable whose distribution was same for all customers. He performed steady state analysis. Haight has analyzed the queue where the individual customer upon arrival measures the queue by its length. Ancker and Gafarian (1963) considered some Queuing Problems with Balking (refusing to join the queue) and Reneging(leaving the queue after entering). They derived steady state probabilities, mean number in queue and system, the probabilities of balking, waiting, reneging, and acquiring service, the customer loss rate, the distribution and mean value of time in queue for customers who acquire service, and the corresponding results for those who renege. Shanthikumar (1988) proved that the queue size decomposition holds even for the M/G/1 models with bulk arrival, reneging, balking etc. Abou-El-Ata (1991) considered M/M/1/N with reneging and general balk functions. The discipline is the classical one FIFO. The probability there are n units in the system, and also some measures of effectiveness are deduced in the steady-state case. Abou-El-Ata and Shawky(1992) considered the single-server Markovian overflow queue with balking, reneging and an additional server for longer queues. They derived an analytical explicit solution of the moments of a system of an overflow queue with balking, reneging and an additional server for longer queues. 36

20 Madan studied a single server queueing system in which arrivals follow a compound Poisson process and the service times of customers have exponential distribution. The system is subject to server vacations occurring randomly in time and in addition the impatient customers resort to balking and reneging during the server's vacation period. The steady state probability generating functions have been obtained explicitly. Various particular cases of interest have also been derived. Ali Movaghar (1998) presented an analytic method for the analysis of queues with statedependent Poisson arrival process, exponential service times, multiple servers, FCFS service discipline, and general Customer impatience. Closed-form solutions are derived for the steady state probabilities of the state process and some important modeling variables and parameters. The efficacy of their method is illustrated through a numerical example. Santhakumaran and Thangaraj (2000) presented a study concerned with several random processes that occur in M/M/1 feedback queues with impatient customers. They have initiated a study on a queueing model with a pair of instantaneous independent Bernoulli feedback processes associated with the queue. The stationary distribution of the arrival process has been obtained. Results for particular queues without feedback are deduced. Madhu Jain (2001) considered a multi-server queuing system in which additional servers are allowed for a longer queue to reduce the customer s balking and reneging behaviour. They obtained equilibrium queue size distribution. The expression for expected number of customers in the system in the long run has been obtained. Some other performance measures have also been provided. The numerical results are presented to verify the validity of the proposed analytical method. Kuo-Hsiung Wang and Jau-Chuan Ke (2003) presented a Probabilistic analysis of a repairable system with warm standbys plus balking and reneging. They considered the steady-state availability and the mean time to system failure of a repairable system 37

21 along with balking and reneging.they presented derivations for the steady-state availability, the mean time to system failure and numerical examples. Van Houdt, Lenin et.al (2003) presented an algorithmic procedure to calculate the delay distribution of (im) patient customers in a discrete time D-MAP/PH/1 queue in three different situations, where the service time distribution of a customer depends on his waiting time. They calculated the delay distribution, using matrix analytic methods, without obtaining the steady state probabilities of the queue length. Possible extensions of this method to more general queues and numerical examples that demonstrate the strength of the algorithm are also included. Amy R.ward and Peter W. Glynnu (2005) presented a Diffusion Approximation for a Queue with Balking or Reneging in a single-server queue with a renewal arrival process amount of time. They established that both the workload and queue-length processes in this system can be approximated. Yan Zhang, Dequan Yue et. al(2005) presented an analysis for an M/M/1/N queueing system with balking, reneging and server vacations. It is assumed that the server has a multiple vacation. By using the Markov process method; they first developed the equations of the steady state probabilities. Then, they derived the matrix form solution of the steady-state probabilities and gave some performance measures of the system. Jau-Chuan Ke (2005) studied the operating characteristics of an M x /G/1 queueing system under a variant vacation policy, where the server leaves for a vacation as soon as the system is empty and balking. They derived the system size distribution at different points in time, as well as the waiting time distribution in the queue. Finally, important system characteristics are also been derived along with some numerical illustration. Liqiang Liu and Vidyadhar G. Kulkarni (2006) considered an M/PH/1 queue with balking based on the workload. An arriving customer joins the queue and stays until served only if the system workload is below a fixed level at the time of arrival. They derived a differential equation for Phase type service time distribution and also solved it 38

22 explicitly, with Erlang, Hyper exponential and Exponential distributions as special cases. They have illustrated the results with numerical examples. Jaejin Jang et.al (2007) estimated the mean waiting time of a customer subject to balking. This paper is aimed to help the design of manufacturing and service systems, being specific to the types of information available. And also presented procedures to estimate the waiting time when customers balk if the anticipated waiting time is larger than a specific value (balking limit). Antonis Economou and Spyridoula Kanta (2008) considered the single server Markovian queue and they assumed that arriving customers decide whether to enter the system or balk based on a natural reward cost structure, which incorporates their desire for service as well as their unwillingness to wait. They examined customers behaviour under the various levels of information regarding the system state and identified equilibrium threshold strategies and also studied the corresponding social and profit maximization problems. Artalejo.J.R and Pla.V (2009) studied the impact of customer balking, impatience and retrials in telecommunication systems. Simple approximations based on truncation and generalized truncations are provided. These approximations are compared according to different criteria for several selected scenarios and are used to evaluate the optimal ratio between the number of available channels and the number of waiting positions in an application to call center management. Yang Woo Shin and Taek Sik Choo (2008) studied queueing with impatient customers and retrials. They considered a queue with balking, reneging and retrials. The number of customers in orbit and service facility is described by a Markov chain on twodimensional lattice space. An algorithm for the stationary distribution of the Markov chain is derived and some numerical results are also presented. 39

23 EL Sherbiny A. A. (2008) studied reneging, balking in batch arrival queues. They derived the solution of the non-truncated queue with reneging, balking, state-dependent and an additional server for longer queues. He has deducted some special cases also. R.O. Al-Seedy et.al (2009) derived the transient solution of the M/M/c queue with balking and reneging. They assumed that arriving customers balk with a fixed probability and renege according to a negative exponential distribution. They used the generating function technique to obtain the transient solution of system which resulted in a simple differential equation. The probabilities are also obtained. M. S. El-Paoumy and M. M. Ismail (2009) studied a Truncated Erlang Queuing System with Bulk Arrivals, Balking and Reneging. They derived the analytical solution of the queue. Recurrence relations connecting the various probabilities introduced are found. Some measures of effectiveness and some special cases are also obtained. Chia-Huang Wu and Jau-Chuan Ke(2010) considered an infinite capacity M/M/c queueing system with c unreliable servers, in which the customers may balk (do not enter) and renege (leave the queue after entering). System performance measures are explicitly derived in terms of computable forms. A cost model is derived to determine the optimal values of the number of servers, service rate and repair rate simultaneously at the minimal total expected cost per unit time. The parameter optimization is illustrated numerically by the Quasi-Newton method. Kuo-Hsiung Wang, Yuh-Ching Liou et.al (2011) presented Cost optimization and Sensitivity Analysis of the Machine Repair Problem with Variable Servers and Balking. They also calculated various system performance measures, such as the expected number of failed machines, the expected number of operating machines, the expected number of busy and idle servers, average balking rate, machine availability, operative utilization, and so on. Antonis Economoua et.al (2012) presented Optimal balking strategies in single-server queues with general service and vacation times. They identified equilibrium strategies 40

24 and socially optimal strategies under two distinct information assumptions. They examined the influence of the information level on the customers strategic response and we compare the resulting equilibrium. Amit Choudhury and Pallabi Medhi (2012) presented a simple analysis of customers impatience in multi-server queues. They assumed that a customer who arrives at the queuing system gets to know the state of the system. Consequently, both balking and reneging are taken as function of system state. Explicit closed form expressions of a number of performance measures are presented. A numerical example is presented to demonstrate the results. Monita Baruah, Kailash C. Madan et.al (2012) studied Balking and Re-service in a Vacation Queue with Batch Arrival and Two Types of Heterogeneous Service. They added the concept of balking and re-service in this study. They have derived the steady state queue size distribution at random epoch and some particular cases have been developed and compared with known results. Kangzhou Wang, Zhibin Jiang et.al (2012) observed the manufacturing system with customer balking behaviour They found the optimal production policy that minimizes the system cost. The effects of system parameters on the optimal base-stock level are analytically investigated, and the impact of customer balking behaviour on the system is illustrated by numerical example in which linear balking function is employed. Rakesh Kumar and Sumeet Kumar Sharma (2012) studied a Queueing Model with Retention of Reneged Customers and Balking. The steady state solution of the model has been obtained and some performance measures have also been computed. The sensitivity analysis of the model has been carried out. The effect of probability of retention on the average system size has also been studied Charan Jeet Singh, Madhu Jain et.al (2012) investigated single server queueing system with varying arrival rates in different states. Various performance indices, such as expected number of units in the queue and in the system, average waiting time, etc., are 41

25 obtained explicitly. Some special cases are also deduced and the numerical illustrations are provided to carry out the sensitivity analysis. Sherif I. Ammar, Mahmoud M. Helan et.al (2013) studied the busy period of an M/M/1 queue with balking and reneging. They derived the transient solution for an M/M/c queueing system through a generating function technique.they further illustrated how this technique can be used to obtain the busy period density function of queue with balking and reneging. Finally, they also presented numerical calculations. G.Ayyappan and S.Syamala (2013) considered a queueing model, wherein the customers are arriving as batches following compound Poisson process. In this model, the customer behaviour balking is considered in both the busy time and server vacation time of the system. They obtained the time dependent solution and the corresponding steady state solutions. Also, they derived the performance measures, the mean queue size and the average waiting time explicitly. 42