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The M/M/ Repairable Queueing System with Variable Breakdown Rates
This paper considers the M/M/ repairable queuing system. The customers' arrival is a Poisson process. The servers are subject to breakdown according to Poisson processes with different rates in idle time and busy time, respectively. The breakdown servers are repaired by repairmen, and the repair time is an exponential distribution. Using probability generating function and transform method, we obtain the steady-state probabilities of the system states, the steady-state availability of the servers, and the mean queueing length of the model.
In queuing researches, many researchers have studied the queuing system with repairable servers. Most of the works of the repairable queuing system deal with the single-server models [1–9]. The works about multiserver repairable systems is not sufficient. Mitrany and Avi-Itzhak  analyzed the model with units of servers and the same amount of repairmen, they obtained the steady-state mean queuing length of customers. Neuts and Lucantoni  studied the model with units of repairable servers and () repairmen by matrix analysis method and obtained the steady-state properties of the model.
In recent years, many flexible policies have been introduced to the repairable systems. Gray et al.  studied the model with a single server which may take a vacation in idle times and may breakdown in busy times; they obtained the mean queue length. Altman and Yechiali  presented a comprehensive analysis of the M/M/1 and M/G/1 queues, as well as of the M/M/ queue with server vacations; they obtained various closed-form results for the probability generating function (PGF) of the number of the customers. Zhang and Hou  analyzed an M/G/1 queue with working vacations and vacation interruptions; they obtained the queue length distribution and steady-state service status probability. Yang et al.  analyzed an M/G/1 queuing system with second optional service, server breakdowns, and general startup times under -policy, they obtained the explicit closed-form expression of the joint optimum threshold values of at the minimum cost. Chang et al.  studied the optimal management problem of a finite capacity M/H2/1 queuing system, where the unreliable server operates -policy, a cost model is developed to determine the optimal capacity , the optimal threshold , the optimal setup rate, and the optimal repair rate at a minimum cost. Wang  used a quasi-birth-and-death (QBD) modeling approach to model queuing-inventory systems with a single removable server, performance measures are obtained by using both hybrid and standard procedures; an optimal control policy is proposed and verified through numerical studies.
The most works of repairable queuing system assumed that the server breakdown rate is constant, but the breakdown rate of a server may be variable in a real system. It is well known that many kinds of machine are easy to breakdown at their busy times, and some equipments may be easy to fail after a long idle period. For example, the tires of the truck prefer to breakdown when the truck is running on the road. On the other hand, the storage battery in an automobile may not work if the automobile is idle for long period. For the actual demands of the above cases, we study a multiserver repairable queuing system in this paper, and assume that the unreliable servers have different breakdown rates in their busy times and idle times, respectively.
The rest of this paper is organized as follows. Section 2 describes the model and gives the balance equations. Section 3 presents the equations of PGF. The steady-state availability of the system is derived in Section 4. The steady-state probabilities of the system states and mean queuing length are obtained in Section 5. Case analysis is given in Section 6. Section 7 presents the conclusions.
2. Model Description
The model characteristics are as follows. (1)There are units of identical servers in the system. The servers are subject to breakdown according to Poisson processes with different rates which are in idle times and in busy times, respectively. (2)Customers arrive according to a Poisson process with rate . The service discipline is first come first served (FCFS). The service which is interrupted by a server breakdown will become the first one of the queue of customers. The service time distribution is an exponential distribution with parameter . (3)There are reliable repairmen to maintain the unreliable servers. The repair discipline is first come first repaired (FCFR). The repair time distribution is an exponential distributions with parameter . A server is as good as a new one after repair.
We define the number of available servers in the system at the moment , the number of customers in the system at the moment .The stochastic process is a two-dimensional Markov process which is called quasi-birth-and-death (QBD) process  with state space .
Let denote the probability that the system is in a state of at the moment , and denote the steady-state probability of , then we have Assuming that the system is positive recurrent, the balance equations are as follows:
Here, we give the derivation of the second equation of (2). Since the process is a vector Markov process of continuous time, we write the equations of the state of by considering the transitions occurring between the moments and as follows: then we have Letting in (5), we have If the system is positive recurrent, we have the formulas . Letting in (6), we obtain the second equation of (2). The derivations of other formulas in (2) are similar.
3. Equations of Probability Generating Functions
The PGFs of the number of customers are defined as follows: Then where is the steady-state probability that the number of the available servers of the system is . Hence,
Multiplying the two sides of every equation of (2) by , and summing over for every , we obtain
We give some explanations of (10), the first equation of (2) multiplied by , we get The second equation of (2) multiplied by , we get Summing (11) and (12) over and using (7), we obtain the first equation of (10). The other equations of (10) are obtained in the same way.
In order to simplify (10), we defined the following notations: Using the above notations, (10) is rewritten as follows: Using Cramer’s rule we obtain where denotes the determinant of , and is a matrix obtained by replacing the th column of with . In (17), the functions of are continuous and bounded in the interval , so the equations in (17) are valid in the interval no matter or not.
4. Steady-State Availability
In this section, we discuss the steady-state availabilities .
Letting in (10) we obtain The equations of (18) are simplified to independent equations, joined with (9), we have Using (2), are reduced to which will be solved in Section 5. Given , we obtain by solving (19), then the steady-state availability of the system is as follows:
5. Steady-State Probabilities of the System States and Mean Queuing Length
5.1. The Roots of in the Interval
In order to get the steady-state probabilities , we need all independent linear equations. Further, for getting the linear equations of we need the roots of in the interval , so we discuss the roots of as follows.
We define the following notations: where is formed by the last rows and last columns of . According to (14) we have The properties of and the necessary proofs are as follows. (a) has no roots. (b) and have no common roots in the interval .
Proof. Suppose that (b) is not true and is a common roots of and , then due to (24). Similarly, , so we get which contradicts the statement (a).
(c) If is a positive root of , then and are opposite in sign. This property readily follows from (24). (d) , .
Proof. Substituting into , we get , , and .
(e) , and .
Proof. The first row of has a common factor , so . The sum of every column of has a common factor . Replacing every element of the last row of with the sum of the corresponding column and extracting the common factor , is written as follows:
If we define
Proof. From the definitions of , we got and , so we get this property from (24).
Proof. It is since the highest power term of is and the sign of is determined by its highest power term.
Theorem 1. If , the polynomial has exactly distinct roots in the interval .
Proof. Since , is a 2-power polynomial of . Further, we find that and , so has two distinct roots which are denoted by and .
With the fact that and are roots of , and , according to the property (c) or (24), we get and .
is a 4-power polynomial of . From the properties (c), (d), (f), and (g), we find that has one and only one root in each interval of , , , and .
So on, we find that is a -power polynomial of , it has distinct roots in the interval and distinct roots in the interval . We denote the roots of by orderly.
From the properties (c), (d), (e), and (f), we find that has one and only one root in each interval , all of them are distinct roots of .
Since and , it has a real number (>0) satisfies and . On the other hand, from (c) and (d) we get then or , so has at least one root in the interval .
From (28), we get . From the property (g), we get . So we know that has at least one root in the interval .
From the properties (e) and (f), we know that 0 and 1 are roots of .
In conclusion, is a -power polynomial of , it has distinct roots at most. Now we make certain all roots of and find that it has distinct roots in the interval .
From the proof, we find that the distinct roots in the interval of are also the roots of .
5.2. Steady-State Probabilities
Assuming that the system parameters meet . Letting denote the roots of in the interval . Substituting in (17), we obtain a set of linear equations about the steady-state probabilities of , but these equations are similar to each other. However, the equations belong to different are independent mutually, so we obtain independent equations by the different roots of , respectively.
In the following, we discuss about the th-independent linear equation of . Similar to (27), is written as follows: where Substituting (27) and (29) into (17), we obtain Substituting in (31), we obtain From (19), we know that can be expressed by , so (32) are linear equations of , but they are similar to each other. However, every equation of (32) is independent with the linear equations obtained by the roots of in the interval . Then all independent linear equations of are as follows: Further, from (29), (33) is equivalent to the follows:
5.3. Mean Queuing Length
6. Case Analysis
We analyze the case of and in this section. According to the above discussion, the determinant (or ) of this case has only one root in the interval . The notations of this case are as follows: then and is equivalent to or The left of (40) is the mean service quantities that all customers need per unit time. The right of (40) is the mean service quantities that the two servers provide per unit time. So (40) is the necessary and sufficient condition of recurrence of the system.
Equations (19) in this case are as follows:
Using (42), we obtain (, ).
For the mean queuing lengths, we have Using (36), we obtain the mean queuing lengths is as follows:
Numerical Example. Letting , , , , , , and , we have The roots of are as follows: and only in the interval .
We find that the mean queuing length () decreases with the increasing of the parameter in Table 1, it is because of the greater service rate the less customers in the system. Furthermore, we find that the availability () increases with the increasing of the parameter , where (the cases: ; on the contrary, the availability decreases with the increasing of the parameter , where (the case: ; otherwise, the availability is constant, where (the case: .
We find that the idle time breakdown rate does not appear in (40). This is because the busy time breakdown rate is at work when the number of the customers is greater than or equal to the number of the available servers, and the criteria of positive recurrence depends on the busy time breakdown rate.
A case analysis is given to illustrate the analysis of this paper, and the numerical results indicate that the variation of breakdown rates has a significant effect on the steady-state availability and steady-state queue length of the system.
This paper is supported by the National Natural Science Foundation of China no. 71071133 and the Natural Science Foundation of Hebei Province no. G2012203136.
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Copyright © 2013 Shengli Lv and Jingbo Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.