This paper is concerned with the analysis of a single server retrial queue with two types of service, Bernoulli vacation and feedback. The server provides two types of service i.e., type 1 service with probability and type 2 service with probability. We assume that the arriving customer who finds the server busy upon arrival leaves the service area and are queued in the orbit in accordance with an FCFS discipline and repeats its request for service after some random time. After completion of type 1 or type 2 service the unsatisfied customer can feedback and joins the tail of the retrial queue with probability or else may depart from the system with probability . Further the server takes vacation under Bernoulli schedule mechanism, i.e., after each service completion the server takes a vacation with probability or with probability waits to serve the next customer. For this queueing model, the steady state distributions of the server state and the number of customers in the orbit are obtained using supplementary variable technique. Finally the average number of customers in the system and average number of customers in the orbit are also obtained.
Keywords:Retrial queues, Two types of service, Steady state, Bernoulli vacation,Bernoulli feedback.
Introduction
Queueing system has a wide range of applications in our day to day life, such as manufacturing industries, airports, traffic-congestion, retail shops, medical shops etc. Queueing system has a prominent place among the modern analytic techniques of Operations Research. From the past decades retrial queueing system plays a vital role in queueing models. Basically it consists of three main components such as arriving customer, orbit (retrial queue) and server. Retrial queueing system is characterized by the phenomenon that an arriving primary customer finds the server busy is supposed to leave the service area and repeat its request for service after some random time. Meanwhile the blocked primary customers are said to be in orbit. These retrial queueing models have applications in many real life situations such as auto-repeat facilities in telephone systems, random access protocols in computer networks, telecommunication networks etc.
The concept of retrial queues was first introduced by Cohen (1957) in which he analyzed the basic problems of telephone traffic theory through the influence of repeated calls. Interested readers may refer the survey papers by Yang and Templeton (1987) and Falin (1990). Earlier studies on retrial queues have considered the classical retrial policy which have been seen detailed in Yang and Templeton (1987) and Falin (1986) where the intervals between successive repeated attempts are exponentially distributed with rate , when the number of customers in the orbit is . However in recent trends there are queueing situations, where the retrial rate is independent of the number of customers in the orbit i.e., the retrial rate is . This retrial policy is called constant retrial policy which was introduced by Fayolle (1986) in the investigation of telephone exchange model. Later Artalejo and Gomez-Corral (1997) have introduced linear retrial policy in which time intervals between successive repeated attempts are exponentially distributed random variables with parameter , where denotes the number of customers in the orbit and denotes the Kronecker delta. Also we have general retrial time policy in which each job in the orbit generates a repeated attempt that are independent of the jobs in orbit and the server state. For detailed study readers may refer Kapyrin (1977) who first introduced the concept of general retrial times in his work. Later Falin (1986) objected this policy but Yang et.al (1994) have established an approximation method for Kapyrin (1977). However in recent years, several retrial models have been analyzed with general retrial times, which have been found in Krishnakumar and Arivudainambi (2002), Krishnakumar et.al (2002) Atencia and Moreno (2004), Atencia and Moreno (2005), Moreno (2004), Wang and Zhao (2007), Choudhury (2009), Ke and Chang (2009), Choudhury and Ke (2012).
Another important and unavoidable concept in this field is server vacation. In many real life queueing situations the server may not be available for a period of time due to break, secondary jobs, maintenance activities, programmed interruptions etc which in-turns termed as vacation. A survey on vacation queues have been analyzed by Doshi (1986). Keilson and Servi (1986) have investigated oscillating random walk models for vacation system with Bernoulli schedules in which he was the first who introduced the concept of Bernoulli vacation. Krishnakumar and Arivudainambi (2002) have investigated an retrial queue with Bernoulli schedule and general retrial times. In this system, the single server takes a Bernoulli vacation i.e., after each service completion, the server takes a vacation with probability , and with probability , it waits to serve the next customer. Atencia and Moreno (2005) have analyzed a single server retrial queue with general retrial times and Bernoulli schedule in which he considered the retrial group (orbit) in accordance with an FCFS discipline. However recently feedback concept received a special attention in the field of queueing models and its applications mainly focused on manufacturing side. Kalidass and Kasturi (2013) have investigated a two phase service queue with a finite number of immediate Bernoulli feedback in which if the customer is unsatisfied with his service than he immediately opt for second round of service after his first round of service.
Retrial queues governing feedback policy have been discussed in the literature Jinting and Jianghua (2006), Arivudainambi and Godhandaraman (2012) have analyzed a batch arrival retrial queue with two phases of service, feedback and K optional vacations, Tao et al (2014) have analysed M/M/1 retrial queue with working vacation interruption and feedback under N-policy and Gao and Liu (2014) have investigated a repairable retrial queue with Bernoulli feedback and impatient customers and so on. In this paper we have considered a single server retrial queue with two types of service, Bernoulli feedback and vacation. In section 2 the mathematical model of the system followed by the notations have been derived . Equation governing the system have been discussed in section 3 and in section 4 the solution of the model under steady state is derived. Finally some of the performance measures are obtained in section 5.
Mathematical description
We consider a single server retrial feedback queue with two types of service and Bernoulli vacation. The detailed mathematical description of the model is as follows, Customers arrive to the system according to a Poisson process at the rate . The single server provides two types of service with probability and in which the customers may choose either type 1 with probability or type 2 with probability having distributions and with corresponding density functions and . We assume that there is no waiting space therefore if the arriving primary customer finds the server busy or in vacation then they leaves the service area and joins into the orbit and repeat its request for service after some random time according to FCFS discipline. Inter- retrial times have an arbitrary distribution with corresponding density function . The server takes a Bernoulli vacation (i.e.,) after each service completion with probability the server takes vacation or with probablity retains in the system to serve the next customer. Similarly the vacation time has distribution function with density function . Further the customer takes a Bernoulli feedback (i.e.,) after service completion the unsatisfied customer may join into the tail of the retrial queue with probability or departs the system with probability . We assume that inter-arrival times, service times,retrial times and server vacation times are all mutually independent. is the conditional completion rate of type service given that the elapsed time is , is the conditional completion rate of retrial given that the elapsed time is and similarly is the conditional completion rate of vacation given that the elapsed time is . The state of the system at time can be defined by the Markov process where denotes the number of customer in the orbit at time .
If and , then (t) represents the elapsed retrial time, if , then corresponds to the elapsed time of the customer being served in type 1 service, if , then corresponds to the elapsed time of the customer being served in type 2 service, and if and , then represents the elapsed vacation time at time . Then the conditional completion rates of serive time during type 1 service, service time during type 2 service, retrial times and vacation times are as follows
Here the arrival stream is Poisson it can be shown from Burke’s theorem [6,p. 187-188] that the steady-state probabilities of exist and positive if and only if . From the mean drift , for , we have the conclusion that the term has three components: new arrivals during the busy period of the server when providing type 1 service , new arrivals during the busy period of the server when providing type 2 service , and new arrivals during vacation . Further, is the expected number of orbiting customers who enter service successfully, given that the previous service time leaves j customers in the orbit. For stability, we require that new customers arrive during a service time and vacation time more slowly than orbiting customers seeking service, at the commencement of service. That is .
Equations governing the system
For the Markov process , we define the probability
and the probability densities
where, = Probability that at time , there are customers in the orbit and the elapsed retrial time is . = Probability that at time , there are customers in the orbit excluding one customer in the type of service and the elapsed service time for this customer is . Consequently denotes the probability that at time there are customers in the queue excluding the one who is getting type of service irrespective of the value of . = Probability that there are customers in the orbit and the elapsed vacation time is . = Probability that there are zero customers in the orbit and the server is free.
Based on the above assumptions and notations, our model is governed by the following set of differential difference equations,
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
These equations are solved subject to the following boundary conditions,
(9)
(10)
(11)
(12)
(13)
(14)
(15)
We assume that the initial conditions are, , , and for since initially there are no customer in the system, the server is not under vacation and so the server is idle. The normalizing condition can be stated as
(16)
The steady state solution
Theorem 1
The system of differential difference equations to describe the feedback retrial queue with two types of service and Bernoulli vacation are given in equations (1)-(15) along with the stability condition , the stationary distributions of the number of jobs in the system when the server being idle, busy with type 1 service, busy with type 2 service and on vacations are
Proof
Define the probability generating functions as
Since the stability condition fulfilled (i.e.,) we have and limiting densities for and , for and and for and . Letting in equations (1)-(15), we have
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
subject to the following boundary conditions
(25)
(26)
(27)
(28)
(29)
(30)
(31)
Multiplying the above equations by and summing from 0 to , we obtain the following partial
differential equations
(32)
(33)
(34)
(35)
Solving the above partial differential equations (32)-(35), we have
(36)
(37)
(38)
(39)
Multiplying equation (25) by suitable powers of , summing over from 0 to and then after some
algebraic manipulations we get,
(40)
Similarly multiplying equations (27), (29) and (31) by suitable powers of , we get
(41)
(42)
(43)
Integrating equation (23) with respect to x between the limits 0 to , we obtain
(44)
Multiplying equation (23) by on both sides and integrating with respect to from 0 to , we get
(45)
Similarly multiplying equations (36) - (39) by , , , and integrating over , we have
(46)
(47)
(48)
(49)
and
(50)
Using equations (45), (47), (48) and (49) in (40) and substituting in (50), we obtain the probability generating functions as
From the above equations, is the probability generating function of orbit size when the server is idle, is the probability generating function of the orbit size when the server is busy in type 1 service, is the probability generating function of the orbit size when the server is busy in type 2 service, is the probability generating function of the orbit when the server is on vacation, and is the probability that the server is idle in the system. Hence the theorem.
Performance measures
From the equations (51)-(54), the only unknown is which can be determined by using the normalization condition . Thus by substituting in that equations and applying L’Hopital’s rule wherever necessary, also using some algebraic manipulations, we get
(51)
The probability generating function of the number of customers in the system and in orbit is obtained by as follows:
(i.e.,) we have
and
The expected number of customers in the orbit under steady state condition is obtained by differentiating with respect to and evaluation at equal to 1 also applying L’Hopital’s rule wherever appropriate, we have
(52)
Similarly denote the expected number of customers in the system under steady state and evaluation at by differentiating and equal to 1, we have
(53)
Special cases
In this section we present some of the special cases of our model.
Case 1: If = 0 and then our model can be reduced to retrial queue
with Bernoulli schedules and general retrial times by Krishnakumar and Arivudainambi (2002).
where
Case 2: If 1, = 0 , and then the system reduces to queue with single vacation model by Doshi (1986). Thus the probability generating function of the number of customers in the system becomes
where
Case 3: If 1, = 0, and then the system reduces to queue one
limited service system with single vacation policy by Takagi (1991). Thus the probability generating function
of the number of customers in the system becomes
and the mean number of customer is obtained as,
and mean waiting time in the system as
Case 4: If 1, = 0, and then the system reduces to the classical queueing system by Gross and Harris (2011),i.e, the well-known Pollaczek-Khintchine formula.
Conclusion
We have introduced single server retrial feedback queue with two types of service and Bernoulli vacation, where the server provides two types of service. The customer arriving for service can take either type 1 service with probability or type 2 service with probability under Bernoulli vacation schedule. For this model, explicit expressions were obtained for the probability generating function of the server states, the number of jobs in a system and the orbit were found using the supplementary variable technique. Some performance measures and special cases have been analyzed.
References
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