Abstract
A discrete-time queue with vacations in random environment is analyzed. Using the method of supplementary variable, we give the probability generating function (PGF) of the stationary queue length distribution at arbitrary epoch. The PGF of the stationary sojourn time distribution is also derived. And we present the various performance measures such as mean number of customers in the system, mean length of the type- cycle, and mean time that the system resides in phase . In addition, we show that the queue with vacations in random environment can be approximated by its discrete-time counterpart. Finally, we present some special cases of the model and numerical examples.
1. Introduction
During the last three decades queueing systems with vacations have been largely studied; see, for example, the surveys (Doshi [1]) and the monographs (Takagi [2] and Tian and Zhang [3]). Queueing systems with vacations are characterized by the feature that each time a busy period ends and the system becomes empty, the server starts a vacation of random length of time. These queueing models have been applied in many fields such as production systems and computer and communication systems. Doshi [1] gives a large number of examples. See also Takagi [2].
Due to the applicability in the field of computer and communication systems in which time is slotted, discrete-time queueing models have been widely used over the past several years. For a detailed discussion and applications of discrete-time queues, see, for example, Hunter [4], Bruneel and Kim [5], and Woodward [6]. The first work on discrete-time queues is due to Meisling [7]. Since then, parallel to the continuous-time vacation queues, discrete-time queues have been intensively studied by many researchers. The early results of discrete-time queues can be found in the book of Hunter [4]. Subsequently, Takagi [8] provided a detailed analysis on queues with a variety of vacation policies. The work about a queue with multiple adaptive vacations was considered by Zhang and Tian [9]. Alfa [10] discussed a model with nonexhaustive service in which both vacation time and service time follow phase type distributions. Recently, Atencia and Moreno [11] studied a discrete-time retrial queue, where the server is subject to starting failures. Li and Tian [12] considered a queue with working vacations and vacation interruption. Using the matrix-analytic method, Tao et al. [13] analyzed the queue with Bernoulli-schedule-controlled vacation and vacation interruption. Vijaya Laxmi et al. [14] analyzed a discrete-time working vacation queue with balking.
In most of the literature on queueing theory, both arrival rate and service rate are homogeneous. But, in real life, the parameters in the queueing models may not be constant; they may change with changes of the environment. Yechiali and Naor [15] study a two-level modification of the queueing model, where the rate of arrival and the service capacity are subject to Poisson alternations. Their work is considered to be the first systematic work on queueing system in random environment. The queue in a two-phase random environment was studied by Neuts [16], Boxma and Kurkova [17], Huang and Lee [18], and so on. Recently, Cordeiro and Kharoufeh [19] discussed an unreliable retrial queue whose arrival, service, failure, repair, and retrial rates are all modulated by an exogenous random environment. B. Kim and J. Kim [20] considered a single server queue with Markov modulated service rates and impatient customers. An queue in random environment with disasters was investigated by Paz and Yechiali [21] and Jiang et al. [22] extended their work to queue.
To the best of our knowledge, there has been no research on the discrete-time queueing system with vacations in random environment. However, in many practical applications like production, these queueing systems may be utilized. For example, in production system, if there is no workload to be processed, the server has a vacation period. When it returns from vacation, if there are one or more customers in the system, it continues to serve customers until the system becomes empty. But the new service rate may change with the changes of the customers’ arrival rate, environmental conditions, and operator experience. This motivated us to study the queue with vacations in random environment in this paper.
The rest of the paper is organized as follows. Section 2 is devoted to the model description. Section 3 obtains the probability generating function of the steady-state queue size distribution at an arbitrary epoch. Section 4 presents the various performance measures such as mean number of customers in the system, mean length of the type- cycle, and PGF of the steady-state sojourn time distribution. Section 5 gives the relationship between the discrete-time system and its continuous-time counterpart. Some special cases are presented in Section 6. Numerical results are presented in Section 7 followed by conclusions in Section 8.
2. Model Description
We consider a discrete-time vacation queue operating in random environment where the time axis is segmented into slots of equal length. It is assumed that the time axis is marked by and all queueing activities occur only at the slot boundaries. We consider late arrival system (LAS) policy in our queueing system. That is, we assume that the arrivals occur in , the departures occur in , and the vacation completion occurs just at the instant .
When the system operates in phase , , customers arrive according to geometrical process with rate , where is the probability that a customer arrives in a slot, and the service times are independent and identically distributed according to a general distribution with probability generating function and mean . Customers are served according to the first-come, first-served (FCFS) discipline. Each time a busy period ends and the system becomes empty, the server leaves for a vacation of random length , which is geometrically distributed with parameter , causing the system to move to vacation phase , where is the probability that vacation ends in a slot. In vacation phase , the arrivals occur according to a geometrical process of rate . When the server returns, if it finds no customer waiting, it goes on another vacation. Otherwise, the system moves from the vacation phase to some operative phase with probability , , where and . The interarrival times, the individual service, and vacation durations are assumed to be mutually independent. Throughout the rest of the paper, we denote .
3. Steady-State Queue Size Distribution
In operative phase , , it is easily seen that the system acts as the classical queue with geometrical arrival rate and service rate . So as long as , , this system that we consider is stable. Assume that the condition for stability of the system is fulfilled. Next, we analyze the steady-state queue size distribution.
At time , the system can be described by the process , where denotes the number of customers in the system and represents the phase in which the system operates. If and , , represents the remaining service time of the customer currently being served during phase . Then is a Markov process with the state space expressed as
Since we are interested in the stationary behavior of the system, define
The balance equations for the stationary distribution of the system arewhere is Kronecker’s delta.
To solve the above equations (3)–(5) let us define the following PGFs: From (4), we obtainand then, we haveMultiplying both sizes of (5) by and then taking summation over all possible values of , we getAgain multiplying both sides of (9) by and then taking summation over all possible values of , we getwhere .
Setting in (10), we getSetting in (11), we getSubstituting (12) into (11), we haveSubstituting (12) and (13) into (10), we obtainSetting in (14), we getThen from (8) and (15), we get the distribution of the number of customers in the system having PGF
Putting , . Let in (15); we obtain by L’Hospital’s RulewhereSubstituting (18) into (17), we getFrom normalizing condition that can be written aswe get on simplificationwhere
Now we summarize the results of this section in the following theorem.
Theorem 1. If the stationary condition , , holds, the PGF of the distribution of the number of customers in the system is given bywhere is given by (22).
4. Performance Measures
In this section, some important performance measures of the system will be provided.
4.1. Mean Number of Customers in the System
The mean number of customers in the system is obtained aswhere , .
After some calculations we get where is the nd moment of the service time. Substituting the above results into (24), we have
4.2. Mean Time That the System Resides in Phase 0
We denote as the interarrival time during a vacation period, which has geometrical distribution with parameter . Then, the probability that there is no customer arrival during a vacation period is given by that is to say, the probability that at least one customer arrives during a vacation period is . We can easily find that the number of vacations in phase is geometrically distributed with parameter and the mean time of each vacation is . Hence, the mean time that the system resides in phase , denoted by , is given bywhich is just the product of the mean number of vacations in phase and the mean time of each vacation.
4.3. Mean Number of Customers in the System at the End of Phase 0
Let be the probability that customers arrive during a vacation , and let represent the probability that customers arrive during phase . Then Then, the mean number of customers in the system at the end of phase , denoted by , is given bywhich is the product of the mean time that the system resides in phase , that is, , and the mean number of customers that arrive to the system per unit time during phase , that is, .
4.4. Mean Time That the System Resides in Phase
We denote as the mean time that the system resides in phase . Thenwhere is the mean busy period that one customer induces in phase . Actually, the mean time that the system resides in phase is the product of the mean number of customers in the system at the end of phase and the mean busy period that one customer induces in phase .
4.5. Probability That the System Resides in Phase 0
From (8) and (21), the probability that the system resides in phase is given byIntuitively, in the time interval between two consecutive epochs at which phase begins, the probability that the system resides in phase is equal to the proportion of time the system resides in phase . Then, we havewhere is the mean time that the system resides in vacation phase and is the mean time that the system resides in operative phase.
4.6. Probability That the System Resides in Phase
From (19) and (21), the probability that the system resides in phase is given bywhich can be intuitively explained in a way similar to the one described above for the probability that the system resides in phase .
4.7. Mean Length of the Type- Cycle
The length of type- cycle, denoted by , is the length of time from the beginning of the last phase to the beginning of the next phase , . Then, there are types of cycles. From (28) and (31), we can easily get the mean length of the type- cycle, denoted by , as equal to . Now, we calculate the mean length of the type- cycle, denoted by , .
We can easily find that the times of the system visiting phase during type- cycle are distributed with geometric distribution with parameter , and the mean length of type- cycle during which the system visits phase 0 times is Then, the mean length of the type- cycle is given by
4.8. PGF of the Stationary Sojourn Time Distribution of an Arbitrary Customer
The stationary sojourn times of an arbitrary customer, a customer who arrives in the state and a customer who arrives in the state , , respectively, are denoted by , , and , with their respective PGFs , , and . We also define and , respectively, as the sum of the customers’ service times and its PGF, where .
When a customer arrives in state , , his sojourn time until departure is the sum of the remaining vacation time and the customers’ service times. Because the vacation time is distributed with a geometrical distribution, the remaining vacation time is identically distributed with vacation time. Then, we havewhere is the PGF of the vacation time distribution and .
When a customer arrives in state , , , , the PGF of the customer’s stationary sojourn time until departure is given by
Hence, we have the PGF of the stationary sojourn time distribution of an arbitrary customer as follows:
5. Relation to the Continuous-Time System
In this section, we analyze the relationship between the discrete-time system and its continuous-time counterpart. The continuous-time system can be approximated by the discrete-time system. If we assume that time is slotted into intervals of constant length , when goes to zero, the approximation tends to the exact value.
We consider the continuous-time queue with vacations in multiphase random environment. In phase , , customers arrive according to a Poisson stream with rate and service times are independent and identically distributed with general distribution function , LST , and finite means . In phase , customers arrive according to a Poisson stream with rate . Vacation times are independent exponential random variables with parameter . If we assume time is slotted into intervals of constant length , the continuous-time system can be approximated by a discrete-time system for which where is sufficiently small so that , , , and are probabilities.
Using the same technique with Yang and Li [23], it is not difficult to show that is the probability generating function of the number of customers in the queue with vacations in multiphase random environment.
From (8), we haveSince we obtainSince we get
From (43) and (45), we obtain the PGF of the number of customers in the queue with vacations in multiphase random environment as follows:
6. Special Cases
6.1. The Queue with Vacations
When the system is homogeneous, that is, and , the model translates into a regular queue with multiple vacations. From (23), reduces towhich is the PGF of the number of customers in the regular queue with multiple vacations, where is the PGF of the service time distribution, . Expression (47) is in agreement with that in Tian and Zhang [3].
6.2. The Queue with Vacations in Random Environment
Suppose that the service times in phase , , follow the geometric distribution with finite means in the model; then in (23), and (26) yieldswhich is the mean number of customers in the queue with vacations in random environment.
7. Numerical Examples
To observe the effect of the different parameters on the main performance measures, we present some numerical examples in this section. We consider the queue with vacations in random environment, and we assume ; that is to say, the system has three phases. We set , , , , , , , and , unless they are considered as variables or their values are given in the respective figures.
In Figure 1, we study the influence of on for different vacation rate . As intuition tells us, the mean number of customers in the system increases with increasing values of for any . On the other hand, for a fixed , decreases with the increase of . Intuitively, as increases, the mean time of vacation decreases; then there are fewer customers in the system. And the behavior of with respect to is displayed in Figure 2. As to be expected, when increases the mean number of customers in the system decreases for any . If is fixed, decreases with the increase of .
The influence of arrival rate on the probability that the system is empty is illustrated in Figure 3. It is observed that decreases as increases, which also agrees with the intuitive expectations. In addition, for a fixed , increases with the increase of . Figure 4 describes the influence of on the probability that the system is empty . increases with the increase of , and, for a fixed , increases with the increase of , as we expected.
8. Conclusion
In this paper, we consider a queue with vacations in random environment. For this model, we derive the probability generating function of the number of customers in the system. Various system performance characteristics are obtained. And we show that the queue with vacations in random environment can be approximated by its discrete-time counterpart. We also perform some numerical examples to demonstrate the effect of various parameters on the performance characteristics.
Competing Interests
The authors declare that they have no competing interests.