Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2013, Article ID 173407, 10 pages http://dx.doi.org/10.1155/2013/173407 Research Article The M/M/đđ Repairable Queueing System with Variable Breakdown Rates Shengli Lv1 and Jingbo Li2 1 2 School of Science, Yanshan University, Qinhuangdao 066004, China College of Mathematics and Information, Hebei Normal University of Science and Technology, Qinhuangdao 066004, China Correspondence should be addressed to Shengli Lv; qhdddlsl@163.com Received 6 October 2012; Accepted 31 December 2012 Academic Editor: Xiaochen Sun Copyright © 2013 S. Lv and J. Li. îĸis 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. îĸis paper considers the M/M/đđ repairable queuing system. îĸe customers’ arrival is a Poisson process. îĸe servers are subject to breakdown according to Poisson processes with diīŦerent rates in idle time and busy time, respectively. îĸe 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. 1. Introduction 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]. îĸe works about multiserver repairable systems is not suīŦcient. Mitrany and Avi-Itzhak [10] analyzed the model with đđ units of servers and the same amount of repairmen, they obtained the steady-state mean queuing length of customers. Neuts and Lucantoni [11] studied the model with đđ units of repairable servers and đđ (0 ≤ đđ đ đđ) repairmen by matrix analysis method and obtained the steady-state properties of the model. In recent years, many īŋŊexible policies have been introduced to the repairable systems. Gray et al. [5] 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 [12] presented a comprehensive analysis of the M/M/1 and M/G/1 queues, as well as of the M/M/đđ 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 [6] 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. [7] analyzed an M/G/1 queuing system with second optional service, server breakdowns, and general startup times under (đđđ đđđ-policy, they obtained the explicit closed-form expression of the joint optimum threshold values of (đđđ đđđ at the minimum cost. Chang et al. [8] studied the optimal management problem of a īŋŊnite capacity M/H2/1 queuing system, where the unreliable server operates đšđšpolicy, a cost model is developed to determine the optimal capacity đžđž, the optimal threshold đšđš, the optimal setup rate, and the optimal repair rate at a minimum cost. Wang [9] 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 veriīŋŊed through numerical studies. îĸe 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 aîer 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 diīŦerent 2 Discrete Dynamics in Nature and Society breakdown rates in their busy times and idle times, respectively. îĸe 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. îĸe steady-state availability of the system is derived in Section 4. îĸe steadystate 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. ķļĄķļĄđđ đ đđđđ đ đđđđ1 ķļąķļą đđđđđđ = đđđđđđđđđđđđ + đđđđđđđđ + (đđ đđ) đđ1 đđđđđđđđ , 0 < đđ đđđđđđđ đđ đđđ đđđđđđđ ķļĄķļĄđđ đ đđđđ đ đđđđ đ ķļĄķļĄđđ đ đđķļąķļą đđ1 +đđđđ2 ķļąķļą = đđđđđđđđđđđ + đđđđđđđđđđđđđ + ķļĄķļĄđđđđķļąķļą đđđđđđđđđđđ + ķļĄķļĄķļĄķļĄđđ đđđ đđķļąķļą đđ1 +đđđđ2 ķļąķļą đđđđđđđđđ , đđđđđđđ ķļĄķļĄđđ đ đđđđ đ đđđđ đ đđđđ2 ķļąķļą 2. Model Description îĸe model characteristics are as follows. (1) îĸere are đđ units of identical servers in the system. îĸe servers are subject to breakdown according to Poisson processes with diīŦerent rates which are đđ1 in idle times and đđ2 in busy times, respectively. (2) Customers arrive according to a Poisson process with rate đđ. îĸe service discipline is īŋŊrst come īŋŊrst served (FCFS). îĸe service which is interrupted by a server breakdown will become the īŋŊrst one of the queue of customers. îĸe service time distribution is an exponential distribution with parameter đđ. (3) îĸere are đđ đđ đ đđ đ đđđ reliable repairmen to maintain the unreliable servers. îĸe repair discipline is īŋŊrst come īŋŊrst repaired (FCFīŋŊ). îĸe repair time distribution is an exponential distributions with parameter đđ. A server is as good as a new one aîer repair. īŋŊe deīŋŊne 0 < đđ đđđđđđđđđ đđ đ đđđ = đđđđđđđđđđđ + đđđđđđđđđđđđđ + đđđđđđđđđđđđđ + ķļĄķļĄđđ1 +đđđđ2 ķļąķļą đđđđđđđđđ , đđđđđđđ ķļĄķļĄđđ đ đđđđ đ đđđđ đ đđđđ2 ķļąķļą 0 < đđ đđđđđđđ đđ đ đđđ = đđđđđđđđđđđ + đđđđđđđđđđđđđ + đđđđđđđđđđđđđ + (đđ đđ) đđ2 đđđđđđđđđ , đđđđđđ ķļĄķļĄđđ đ (đđ đđđ) đđ đđđđđ1 ķļąķļą 0 < đđ đđđđđđđ đđ đ đđđ = (đđ đđđđđ) đđđđđđđđđđ + đđđđđđđđ + (đđ đđ) đđ1 đđđđđđđđ , đđ đđđđđđđ đđđđđđđđ đđđđđđđ ķļĄķļĄđđ đ (đđ đđđ) đđ đđđđđđ ķļĄķļĄđđ đ đđķļąķļą đđ1 +đđđđ2 ķļąķļą = (đđ đđđđđ) đđđđđđđđđđđ + đđđđđđđđđđđ + ķļĄķļĄđđđđķļąķļą đđđđđđđđđđđ + ķļĄķļĄķļĄķļĄđđ đđđ đđķļąķļą đđ1 +đđđđ2 ķļąķļą đđđđđđđđđ , đđ đđđđđđđ đđđđđđđđđđđ đđđđđđ đ the number of available servers in the system at the moment đĄđĄđĄđĄ đĄ đĄđĄđĄt)âŠŊ đđđ, đđđđđđđ ķļĄķļĄđđ đ (đđ đđđ) đđ đđđđđđđđđđ2 ķļąķļą îĸe stochastic process {đđđđđđđ đđđđđđđ đđ đ đđ is a two-dimensional Markov process which is called quasi-birth-and-death (QBD) process [11] with state space {(đđđ đđđđđđ đđ đđđđ đđ đđđ. Let đđđđđđđ (đĄđĄđĄ denote the probability that the system is in a state of (đđđ đđđ at the moment đĄđĄ, and đđđđđđđ denote the steady-state probability of đđđđđđđ (đĄđĄđĄ, then we have đđđđđđđ ķļĄķļĄđđ đ (đđ đđđ) đđ đđđđđđđđđđ2 ķļąķļą đđđđđđđ the number of customers in the system at the moment đĄđĄđĄđĄ đĄ đĄđĄđĄđĄđĄđĄđĄ. lim đđđđđđđ (đĄđĄ) , đđ đđđđđđđ đ đđđđ đđ đđđđđđđ đ đ đđđđđđđ = ķļķļđĄđĄ đĄ đĄ 0, other. (1) Assuming that the system is positive recurrent, the balance equations are as follows: ķļĄķļĄđđ đ đđđđķļąķļą đđ0,0 = đđ1 đđ1,0 , ķļĄķļĄđđ đ đđđđķļąķļą đđ0,đđ = đđđđ0,đđđđ + đđ2 đđ1,đđ ,đđ đ đđ = đđđđđđđđđđđ + (đđ đđđđđ) đđđđđđđđđđđ + đđđđđđđđđđđđđ + ķļĄķļĄđđ1 +đđđđ2 ķļąķļą đđđđđđđđđ , đđ đđđđđđđ đđđđđđđđđ = đđđđđđđđđđđ + (đđ đđđđđ) đđđđđđđđđđđ + đđđđđđđđđđđđđ + (đđ đđ) đđ2 đđđđđđđđđ , đđ đđđđđđđ đđđđđđđđđ đđđđđđ ķļĄķļĄđđ đ đđđđ1 ķļąķļą = đđđđđđđđđđ + đđđđđđđđ , đđđđđđđ ķļĄķļĄđđ đ đđđđ đ ķļĄķļĄđđ đđđķļąķļą đđ1 +đđđđ2 ķļąķļą đđ đđđđ đđ đđđ = đđđđđđđđđđđ + đđđđđđđđđđđ + ķļĄķļĄđđđđķļąķļą đđđđđđđđđđđ , đđ đđđđđđ đđ đđđđ đđđđđđđ ķļĄķļĄđđ đ đđđđ đ đđđđ2 ķļąķļą = đđđđđđđđđđđ + đđđđđđđđđđđ + đđđđđđđđđđđđđ , đđ đđđđ đđ đđđđ (2) Discrete Dynamics in Nature and Society 3 Here, we give the derivation of the second equation of (2). Since the process {đđđđđđđ đđđđđđđ đđ đ đđ is a vector Markov process of continuous time, we write the equations of the state of (0, đđđ by considering the transitions occurring between the moments đĄđĄ and đĄđĄ đĄ đĄđĄđĄđĄđĄđĄđĄ đĄ đĄđĄ as follows: đđ0,đđ (đĄđĄ đĄ đĄđĄđĄ) = đđ0,đđđđ (đĄđĄ) đđđđđđđđ1,đđ (đĄđĄ) đđ2 ΔđĄđĄ then we have + đđ0,đđ (đĄđĄ) ķļĄķļĄ1− ķļĄķļĄđđ đ đđđđķļąķļą ΔđĄđĄķļąķļą + đđ (ΔđĄđĄ) , đđ0,đđ (đĄđĄ đĄ đĄđĄđĄ) − đđ0,đđ (đĄđĄ) đđ0,đđ (đĄđĄ đĄ đĄđĄđĄ) − đđ0,đđ (đĄđĄ) ΔđĄđĄ = đđ0,đđđđ (đĄđĄ) đđđđđđđđ1,đđ (đĄđĄ) đđ2 ΔđĄđĄ (3) − đđ0,đđ (đĄđĄ) ķļĄķļĄđđ đ đđđđķļąķļą + Letting ΔđĄđĄ đĄ đĄ in (5), we have đđ (ΔđĄđĄ) . ΔđĄđĄ đđ0,đđ (đĄđĄ)′ = đđ0,đđđđ (đĄđĄ) đđ đđđ1,đđ (đĄđĄ) đđ2 − đđ0,đđ (đĄđĄ) ķļĄķļĄđđ đ đđđđķļąķļą . (4) îĸen ∞ đēđēđđ (1) = ķĩ ķĩ đđđđđđđ đđđđ (5) đđđđ đđđđ đđđđ − (đđ đđđđđ) đđđđđđđđđđ (đ§đ§) + ķļķļđ§đ§ ķļĄķļĄđđđđ đ đđđđ2 + đđ đ (đđ đđđ) đđķļąķļą − đđđđ2 − đđđđķļķļ đēđēđđ (đ§đ§) (6) đđđđ = ķĩ ķĩ ķļĄķļĄđđ2 − đđ1 ķļąķļą (đđ đđđ) đđđđđđđ đ§đ§đđđđ đđđđ đđ + ķĩ ķĩ ķļĄķļĄđđ1 − đđ2 ķļąķļą (đđ đđđđđ) đđđđđđđđđ đ§đ§đđđđ đđđđ đđđđ + (đ§đ§ đ§đ§) ķĩ ķĩ đđ (đđ đđđ) đđđđđđđ đ§đ§đđ , đđđđ đđ đđđđđđđ đđđ −đđđđđđđđđđ (đ§đ§) + ķļĸķļĸđ§đ§ ķļĄķļĄđđđđđ đđđđ2 + đđķļąķļą − đđđđ2 − đđđđķļ˛ķļ˛ đēđēđđ (đ§đ§) đđ đēđē (đ§đ§) ≡ ķĩ ķĩ đēđēđđ (đ§đ§) , đđđđ (7) 0 ≤ đđ đđđđ |đ§đ§| ≤ 1. (đđ đđđđđđđ đ đđđ) , ķĩ ķĩ đēđēđđ (1) = 1. đđđđ − (đđ đđ) đ§đ§đ§đ§2 đēđēđđđđ (đ§đ§) đđđđ = ķĩ ķĩ ķļĄķļĄđđ2 − đđ1 ķļąķļą ķļąđđ đđđ) đđđđđđđ đ§đ§đđđđ đđđđ đđđđ + (đ§đ§ đ§đ§) ķĩ ķĩ đđ (đđ đđđ) đđđđđđđ đ§đ§đđ . đđđđ (10) We give some explanations of (10), the īŋŊrst equation of (2) multiplied by đ§đ§, we get (8) where đēđēđđ (1) is the steady-state probability that the number of the available servers of the system is đđ. Hence, đđ đđ + ķĩ ķĩ ķļĄķļĄđđ1 − đđ2 ķļąķļą ķļąđđ đđđđđ) đđđđđđđđđ đ§đ§đđđđ + (đ§đ§ đ§đ§) ķĩ ķĩ đđ (đđ đđđ) đđđđđđđ đ§đ§đđ ,0 < đđ đđđđđđđ îĸe īŋŊīŋŊīŋŊs of the number of customers are deīŋŊned as follows: đđđđ − (đđ đđ) đ§đ§đ§đ§2 đēđēđđđđ (đ§đ§) đđđđ 3. Equations of Probability Generating Functions đēđēđđ (đ§đ§) ≡ ķĩ ķĩ đ§đ§đđ đđđđđđđ , −đđđđđđđđđđđđ (đ§đ§) + ķļĸķļĸđ§đ§ ķļĄķļĄđđđđ đ đđđđ2 + đđ đ đđđđķļąķļą − đđđđ2 − đđđđķļ˛ķļ˛ đēđēđđ (đ§đ§) đđđđ If the system is positive recurrent, we have the formulas limđĄđĄ đĄ đĄ đđ0,đđ (đĄđĄđĄ′ = 0 [13]. Letting đĄđĄ đĄ đĄ in (6), we obtain the second equation of (2). îĸe derivations of other formulas in (2) are similar. ∞ đ§đ§ ķļĄķļĄđđ đ đđđđđ đđđđķļąķļą đēđē0 (đ§đ§) − đ§đ§đ§đ§2 đēđē1 (đ§đ§) = đ§đ§ ķļĄķļĄđđ1 − đđ2 ķļąķļą đđ1,0 , = ķĩ ķĩ ķļĄķļĄđđ2 − đđ1 ķļąķļą ķļąđđ đđđ) đđđđđđđ đ§đ§đđđđ − đđ0,đđ (đĄđĄ) ķļĄķļĄđđ đ đđđđķļąķļą ΔđĄđĄ đĄ đĄđĄ (ΔđĄđĄ) , = đđ0,đđđđ (đĄđĄ) đđ đđđ1,đđ (đĄđĄ) đđ2 Multiplying the two sides of every equation of (2) by đ§đ§đđđđ , and summing over đđ đđđ đđđ đđđđđđ for every đđ, we obtain (9) ķļĄķļĄđđ đ đđđđķļąķļą đđ0,0 đ§đ§ đ§đ§đ§1 đđ1,0 đ§đ§đ§ (11) îĸe second equation of (2) multiplied by đ§đ§đđđđ , we get ķļĄķļĄđđ đ đđđđķļąķļą đđ0,đđ đ§đ§đđđđ = đđđđ0,đđđđ đ§đ§đđđđ + đđ2 đđ1,đđ đ§đ§đđđđ , đđ đđđ (12) Summing (11) and (12) over đđ and using (7), we obtain the īŋŊrst equation of (10). îĸe other equations of (10) are obtained in the same way. 4 Discrete Dynamics in Nature and Society In order to simplify (10), we deīŋŊned the following notations: đđđđ (đ§đ§) ≡ ķļĄķļĄđđđđ đ đđđđ2 + đđ đ đđđđķļąķļą đ§đ§ đ§ đ§đ§đ§đ§2 − đđđđđ đđđđ (đ§đ§) ≡ ķļĄķļĄđđđđ đ đđđđ2 + đđ đ (đđđđđ) đđķļąķļą đ§đ§ đ§ đ§đ§đ§đ§2 − đđđđđ đđ đ đđ đđđ đ đđ đđ đđđ đ đđ đ đ đđđ đđ đ đđ đđ đđ đ đ đđ đđđđ đđ0 (đ§đ§) −đđ2 đ§đ§ ķļķļ −đđđđđđđđ1 (đ§đ§) −2đđ2 đ§đ§ ķļķļ âą âą ķļķļ ķļķļ âą ķļķļ ķļķļ ķļķļ đ´đ´ (đ§đ§) ≡ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ âą âą âą −đđđđđđđđđđđđđđđ (đ§đ§) − (đđđđđđđ) đđ2 đ§đ§ (đđđđ) đđđđđđđđđđđđđ (đ§đ§) âą đđ0 (đ§đ§) ≡ đ§đ§ ķļĄķļĄđđ1 −đđ2 ķļąķļą đđ1,0 , đđđđ đđ đđđđđđđđđđ (1) − đđđđ2 đēđēđđ (1) − ķļĄķļĄđđđđđđđđ (1) − (đđ đ đ) đđ2 đēđēđđđđ (1)ķļąķļą đđđđ đđ đđđđ đđđđ đđđđ + (đ§đ§ đ§ đ§) ķĩ ķĩ đđ (đđđđđ) đđđđđđđ đ§đ§đđ , đđ0 (đ§đ§) ķļķļ đđ1 (đ§đ§) ķļ ķļ ķļķļ , đđ (đ§đ§) ≡ ķļķļ ⎠ķļķļđđđđ (đ§đ§)ķļķļ đđđđ đēđē0 (đ§đ§) ķļķļ đēđē1 (đ§đ§) ķļ ķļ ķļķļ . đđ (đ§đ§) ≡ ķļķļ ⎠ķļķļđēđēđđ (đ§đ§)ķļķļ đđ đ đđ đđ đđ đ đ đđđ 0 < đđ đ đđ đđđđ − ķļĄķļĄķļĄđđđđđ) đđđđđđ (1) − (đđ đ đ) đđ2 đēđēđđđđ (1)ķļąķļą (15) (16) đđ đđđđ = ķļĄķļĄđđ1 −đđ2 ķļąķļą ķĩ ķĩ đđđđđđđđđđđđ − ķļĄķļĄđđ1 −đđ2 ķļąķļą ķĩ ķĩ đđđđđđđđđđđđđđđđ , đđđđ đđđđ đđđđđđđđđđđđ đđ đđđđđđđđ (1) −đđđđ2 đēđēđđ (1) = ķļĄķļĄđđ1 −đđ2 ķļąķļą ķĩ ķĩ đđđđđđđđđđđđ . đđđđ Using Cramer’s rule we obtain |đ´đ´ (đ§đ§)| đēđēđđ (đ§đ§) = ķļĄķļĄđ´đ´đđ (đ§đ§)ķļĄķļĄ , đđđđ [đđđ (đđ đ đ)] đđđđđđđđ (1) − đđđđ2 đēđēđđ (1) Using the above notations, (10) is rewritten as follows: đ´đ´ (đ§đ§) đđ (đ§đ§) = đđ (đ§đ§) . đđđđ = ķļĄķļĄđđ1 −đđ2 ķļąķļą ķĩ ķĩ đđđđđđđđđđđđ − ķļĄķļĄđđ1 −đđ2 ķļąķļą ķĩ ķĩ đđđđđđđđđđđđđđđđ , đđđđ (đ§đ§) ≡ ķĩ ķĩ ķļĄķļĄđđ2 −đđ1 ķļąķļą (đđđđđ) đđđđđđđ đ§đ§đđđđ đđđđ −2đđđđđđđđđđ (đ§đ§) −đđđđ2 đ§đ§ −đđđđđđđđ (đ§đ§) ķļķļ đđđđđđ0 (1) −đđ2 đēđē1 (1) = ķļĄķļĄđđ1 −đđ2 ķļąķļą đđ1,0 , + (đ§đ§ đ§ đ§) ķĩ ķĩ đđ (đđ đđđ) đđđđđđđ đ§đ§đđ ,0 < đđ đ đđđ đđđđ âą (14) In this section, we discuss the steady-state availabilities đēđēđđ (1) (đđ đ đđ đđ đđ đ đ đđđ. Letting đ§đ§ đ§đ§ in (10) we obtain + ķĩ ķĩ ķļĄķļĄđđ1 −đđ2 ķļąķļą ķļąđđ đ đ đđđ) đđđđđđđđđ đ§đ§đđđđ đđđđ âą âą 4. Steady-State Availability đđđđ (đ§đ§) ≡ ķĩ ķĩ ķļĄķļĄđđ2 −đđ1 ķļąķļą ķļąđđ đđđ) đđđđđđđ đ§đ§đđđđ đđđđ − (đđđđđđ đ) đđ2 đ§đ§ âą âą ķļ ķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ , ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ (13) (17) where |đ´đ´đ´đ´đ´đ´đ´ denotes the determinant of đ´đ´đ´đ´đ´đ´, and đ´đ´đđ (đ§đ§đ§ is a matrix obtained by replacing the (đđđđđth column of đ´đ´đ´đ´đ´đ´ with đđđđđđ. In (17), the functions of đ§đ§ are continuous and bounded in the interval [0,1], so the equations in (17) are valid in the interval [0,1] no matter |đ´đ´đ´đ´đ´đ´đ´đ´đ´ or not. (18) îĸe (đđđđđ equations of (18) are simpliīŋŊed to đđ independent equations, joined with (9), we have đđđđđđđđ (1) − (đđ đ đ) đđ2 đēđēđđđđ (1) đđđđ = ķļĄķļĄđđ1 −đđ2 ķļąķļą ķĩ ķĩ đđđđđđđđđđđđđđđđ ,0≤ đđ đ đđ đđđđ đđđđ Discrete Dynamics in Nature and Society 5 (đđ đ đđ) đđđđđđ (1) − (đđ đ đ) đđ2 đēđēđđđđ (1) đđđđ = ķļĄķļĄđđ1 − đđ2 ķļąķļą ķĩ ķĩ đđđđđđđđđđđđđđđđ , đđđđ đđ ķĩ ķĩ đēđēđđ (1) = 1. 5. Steady-State Probabilities of the System States and Mean Queuing Length đđ đ đđ đ đđ đ đđ đ đđ đđđđ (19) Using (2), đđđđđđđ (0≤đđđđđđđđđđđđđđđđ are reduced to đđđđđđ (1 ≤đđđđđđ which will be solved in Section 5. Given đđđđđđ (1 ≤đđđđđđ, we obtain đēđēđđ (1) (đđđ đđ đđ đđ đ đ đđđ by solving (19), then the steady-state availability of the system is as follows: 5.1. îģe Roots of |đ´đ´đ´đ´đ´đ´đ´ in the Interval (0, 1). In order to get the steady-state probabilities đđđđđđ (1 ≤đđđđđđ, we need all đđ independent linear equations. Further, for getting the linear equations of đđđđđđ (1 ≤đđđđđđ we need the roots of |đ´đ´đ´đ´đ´đ´đ´ in the interval (0, 1), so we discuss the roots of |đ´đ´đ´đ´đ´đ´đ´ as follows. īŋŊe deīŋŊne the following notations: đđ0 (đ§đ§) ≡ 1, (20) đ´đ´ đ´đ´đ´đ´đ´0 (1) . đđ1 (đ§đ§) ≡ đđđđ (đ§đ§) , (21) đđđđđđ (đ§đ§) −đđđđ2 đ§đ§ ķļĻķļĻ , −đđđđđđđđ (đ§đ§) ⎠⎠(đ§đ§) ≡ ķļĻķļĻ đđ1 (đ§đ§) −2đđ2 đ§đ§ ķļķļ −đđđđđđđđ2 (đ§đ§) −3đđ2 đ§đ§ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ âą âą âą ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ đ§đ§ − đ đđ đđ) đđ −đđđđđđđđ (đ§đ§) (đđ 2 đđđđđđđ ķļķļ , đđđđ (đ§đ§) ≡ ķļķļķļķļ ķļķļ đ§đ§ đđđđđđ − đ đđ đđ) đđ (đđđđ) (đđ 2 ķļķļ ķļķļ đđđđđđđ (đ§đ§) ķļķļ ķļķļ âą âą âą ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ −2đđđđ đđđđđđ (đ§đ§) −đđđđ2 đ§đ§ķļķļ −đđđđđđđđ (đ§đ§) ķļķļ ķļķļ đđđđđđ (đ§đ§) ≡ |đ´đ´ (đ§đ§)| , (23) (0≤đ§đ§đ§ đ§) , where đđđđ (đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§ is formed by the last đđ rows and last đđ columns of |đ´đ´đ´đ´đ´đ´đ´. According to (14) we have 2 đđđđđđ (đ§đ§) = đđđđđđđ (đ§đ§) đđđđ (đ§đ§) − (đđ đ đđđđ) đđ2 đđđđđđ đđđđđđ (đ§đ§) , 1 ≤ đđ đđđđđđ đđđđđđ (đ§đ§) = đđđđđđđ (đ§đ§) đđđđ (đ§đ§) − (đđ đ đđđđ) đđ2 đđđđđđ2 đđđđđđ (đ§đ§) , đđđđđđđđđ (24) îĸe properties of đđđđ (đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§ and the necessary proofs are as follows. (a) đđ0 (đ§đ§đ§ has no roots. (b) đđđđ (đ§đ§đ§ and đđđđđđ (đ§đ§đ§ have no common roots in the interval (0,∞) (đđ đđđđđđđđđđ. (22) Proof. Suppose that (b) is not true and đ§đ§0 (> 0) is a common roots of đđđđ (đ§đ§đ§ and đđđđđđ (đ§đ§đ§, then đđđđđđ (đ§đ§0 ) = 0 due to (24). Similarly, đđđđđđ (đ§đ§0 ) = 0, so we get đđ0 (đ§đ§0 ) = 0 which contradicts the statement (a). (c) If đ§đ§0 is a positive root of đđđđ (đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§, then đđđđđđ (đ§đ§0 ) and đđđđđđ (đ§đ§0 ) are opposite in sign. îĸis property readily follows from (24). (d) đđđđ (1) > 0, đđ đđđđđđđđđđđ. Proof. Substituting đ§đ§đ§đ§ into đđđđ (đ§đ§đ§, we get đđđđ (1) = (đđ đ đđ đđđđđđđ đđ đđđ đ đđđđđđđđđđđđ2 > 0, đđ đđđđđđđđđ, and đđ0 (1) = 1. (e) đđđđđđ (1) = |đ´đ´đ´đ´đ´đ´đ´đ´, and đđđđđđ (0) = |đ´đ´đ´đ´đ´đ´đ´đ´. Proof. îĸe īŋŊrst row of |đ´đ´đ´đ´đ´đ´đ´ has a common factor đ§đ§, so đđđđđđ (0) = |đ´đ´đ´đ´đ´đ´đ´đ´. îĸe sum of every column of |đ´đ´đ´đ´đ´đ´đ´ has a common factor (đ§đ§đ§đ§đ§. Replacing every element of the last row of |đ´đ´đ´đ´đ´đ´đ´ with the sum of the corresponding column and extracting the common factor (đ§đ§đ§đ§đ§, |đ´đ´đ´đ´đ´đ´đ´ is written as follows: 6 Discrete Dynamics in Nature and Society ⯠0 0 đ§ 0 0 ķļ ķļ ķļķļ ķļķļ ⯠0 0 ķļķļ . ķļķļ ⎠ķļķļ ķļķļ 0 0 0 ⯠đđđđđđ (đ§đ§) −đđđđ2 đ§đ§ −đđđđđđđđđ đ đđ đđđđđ đ đđđ đđđđđđ đ (đđ đđ) đđđđđđđđđđđđķļķļ đđ đ đđđđ đ đđđđđđđ2 −đđđđđđ đđ1 (đ§đ§) 0 −đđđđ If we deīŋŊne ķļķļ ķļķļ ķļķļ |đ´đ´ (đ§đ§)| = đ§đ§ (đ§đ§ đ§ đ§) × ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļķļķļ đˇđˇ (đ§đ§) ≡ ķļķļķļķļ ķļķļķļķļ ķļķļ 0 −2đđ2 đ§đ§ đđ2 (đ§đ§) ⯠0 0 đ§ 0 0 ķļķļ ķļķļķļķļ ⯠0 0 ķļķļķļķļ , ⎠ķļķļ 0 0 0 ⯠đđđđđđ (đ§đ§) −đđđđ2 đ§đ§ ķļķļ −đđđđđđđđđ đ đđ đđđđđ đ đđđ đđđđđđ đ (đđ đđ) đđđđđđđđđđđđķļķļ đđ đ đđđđ đ đđđđđđđ2 −đđđđđđ đđ1 (đ§đ§) 0 −đđđđ 0 −2đđ2 đ§đ§ đđ2 (đ§đ§) then |đ´đ´ (đ§đ§)| = đ§đ§ (đ§đ§ đ§ đ§) đˇđˇ (đ§đ§) . (27) (f) Sign[đđđđ (0)] = (−1)đđ (đđ đđđđđđđ đ đđđđ. Proof. From the deīŋŊnitions of đđđđ (đ§đ§đ§đ§đ§đ§đ§đ§đ§ đ§đ§đ§đ§đ§đ§đ§đ§đ§, we got đđ0 (0) = 0 and đđđđ (0) < 0 (đđ đđđđđ đ đđđđ, so we get this property from (24). (g) Sign[đđđđ (+∞)] = (−1)đđ (đđ đđđđđđđ đ đđđđđđ. Proof. It is since the highest power term of đđđđ (đ§đ§đ§ is (−đđđđ2 )đđ (đđ đđđđđđđ đ đđđđđđ and the sign of đđđđ (+∞) is determined by its highest power term. îĸeorem 1. If đˇđˇđˇđˇđˇ đˇ đˇ, the polynomial |đ´đ´đ´đ´đ´đ´đ´ has exactly (đđ đđđ distinct roots in the interval (0, 1). Proof. Since đđ1 (đ§đ§đ§đ§đ§đ§đđ (đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§đ§2 ) + đđđđđđ đđđđ2 − đđđđ, đđ1 (đ§đ§đ§ is a 2-power polynomial of đ§đ§. Further, we īŋŊnd that đđ1 (1) = đđ2 > 0 and đđ1 (0) = −đđđđ, so đđ1 (đ§đ§đ§ has two distinct roots which are denoted by đ§đ§1,1 (0 < đ§đ§1,1 < 1) and đ§đ§1,2 (>1). With the fact that đ§đ§1,1 and đ§đ§1,2 are roots of đđ1 (đ§đ§đ§, and đđ0 (đ§đ§đ§đ§ đ§ đ§đ§, according to the property (c) or (24), we get đđ2 (đ§đ§1,1 ) < 0 and đđ2 (đ§đ§1,2 ) < 0. đđ2 (đ§đ§đ§ is a 4-power polynomial of đ§đ§. From the properties (c), (d), (f), and (g), we īŋŊnd that đđ2 (đ§đ§đ§ has one and only one root in each interval of (0, đ§đ§1,1 ), (đ§đ§1,1 , 1), (1, đ§đ§1,2 ), and (đ§đ§1,2 , +∞). So on, we īŋŊnd that đđđđ (đ§đ§đ§ is a 2đđ-power polynomial of đ§đ§, it has đđ distinct roots in the interval (0, 1) and đđ distinct roots in the interval (1, ∞). We denote the 2đđ roots of đđđđ (đ§đ§đ§ by đ§đ§đđđđđ (đđ đđđđđđđđđđđ orderly. From the properties (c), (d), (e), and (f), we īŋŊnd that |đ´đ´đ´đ´đ´đ´đ´ has one and only one root in each interval (đ§đ§đđđđđ , đ§đ§đđđđđđđ ) (đđ đđđđđđđđđđđđđđđđđđđđđđđđđ, all of them are 2(đđ đđđ distinct roots of |đ´đ´đ´đ´đ´đ´đ´. (25) (26) Since |đ´đ´đ´đ´đ´đ´đ´đ´ and đˇđˇđˇđˇđˇđˇđˇđˇđˇđˇđˇđˇđˇđˇđˇđˇ′đ§đ§đ§đ§ > 0, it has a real number đđ (>0) satisīŋŊes 1+ đđ đđđđđđđđđđ and |đ´đ´đ´đ´đ´đ´đ´đ´đ´đ´đ´. On the other hand, from (c) and (d) we get Sign ķļĄķļĄđđđđđđ ķļĄķļĄđ§đ§đđđđđ ķļąķļąķļąķļą = (−1)đđđđđ , đđ đđđđđđđđđđđđđđđđđ (28) then Sign[đđđđđđ (đ§đ§đđđđđđđ )] = (−1)đđđđđđđ or |đ´đ´đ´đ´đ´đđđđđđđ )| = đđđđđđ (đ§đ§đđđđđđđ ) < 0, so |đ´đ´đ´đ´đ´đ´đ´ has at least one root in the interval (1, đ§đ§đđđđđđđ ). From (28), we get Sign[|đ´đ´đ´đ´đ´đđđđđđđ |] = Sign[đđđđđđ (đ§đ§đđđđđđ )] = (−1)đđđđđđ = (−1)3đđ = (−1)đđ . From the property (g), we get Sign[|đ´đ´đ´đ´đ´đ´đ´đ´đ´ Sign[đđđđđđ (+∞)] = (−1)đđđđ . So we know that |đ´đ´đ´đ´đ´đ´đ´ has at least one root in the interval (đ§đ§đđđđđđ , ∞). From the properties (e) and (f), we know that 0 and 1 are roots of |đ´đ´đ´đ´đ´đ´đ´. In conclusion, |đ´đ´đ´đ´đ´đ´đ´ is a 2(đđ đđđ-power polynomial of đ§đ§, it has 2(đđ đđđ distinct roots at most. Now we make certain all roots of |đ´đ´đ´đ´đ´đ´đ´ and īŋŊnd that it has đđ đđ distinct roots in the interval (0, 1). From the proof, we īŋŊnd that the (đđ đđđ distinct roots in the interval (0, 1) of |đ´đ´đ´đ´đ´đ´đ´ are also the roots of đˇđˇđˇđˇđˇđˇ. 5.2. Steady-State Probabilities. Assuming that the system parameters meet đˇđˇđˇđˇđˇ đˇ đˇ. Letting đ§đ§đđ (đđ đđđđđ đ đđđđđđ denote the roots of |đ´đ´đ´đ´đ´đ´đ´ in the interval (0, 1). Substituting đ§đ§1 in (17), we obtain a set of linear equations about the steadystate probabilities of đđđđđđ (đđ đđđđđđđđđđ, but these equations are similar to each other. However, the equations belong to diīŦerent đ§đ§đđ (đđ đđđđđ đ đđđđđđ are independent mutually, so we obtain (đđ đđđ independent equations by the (đđ đđđ diīŦerent roots of đ§đ§đđ , respectively. In the following, we discuss about the đđth-independent linear equation of đđđđđđ (đđ đđđđđđđđđđ. Similar to (27), |đ´đ´đđ (đ§đ§đ§đ§ is written as follows: ķļĄķļĄđ´đ´đđ (đ§đ§)ķļĄķļĄ = đ§đ§ (đ§đ§ đ§ đ§) đˇđˇđđ (đ§đ§) , đđ đđđđđđđđđđđđ (29) Discrete Dynamics in Nature and Society 7 6. Case Analysis where đˇđˇđđ (đ§đ§) đđđđ đ đđ (1 − đ§đ§) −đđđđđđ 0 ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ = ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ 0 −đđ2 ⯠đđ1 (đ§đ§) ⯠−đđđđđđđ ⎠0 ⯠đđ ķļĄķļĄđđ1 − đđ2 ķļąķļą đđ1,0 đđ1 (đ§đ§) đđ2 (đ§đ§) đđ đđđđđđ (đ§đ§) −đđđđđđđđđđ đđ đ ķĩ ķĩ ķĩ ķĩ đđđđđđđđđđđđđđ đ§đ§đđđđđ đđđđ đđđđ (đđđđđth column ⯠⯠⯠⎠⯠0 0 0 ķļķļ ķļķļ ķļķļ ķļķļ ķļķļ −đđđđ2 đ§đ§ ķļķļ . ķļķļ ķļķļ ⯠−đđđđđđđđđķļķļķļķļ Substituting (27) and (29) into (17), we obtain ķļķļ (30) đˇđˇ (đ§đ§) đēđēđđ (đ§đ§) = đˇđˇđđ (đ§đ§) , đđ đđđđđđđ đ đđđđ (31) đˇđˇ (1) đēđēđđ (1) = đˇđˇđđ (1) , đđ đđđđđđđ đ đđđđ (32) Substituting đ§đ§ đ§đ§ in (31), we obtain From (19), we know that đēđēđđ (1) (đđ đđđđđđđ đ đđđđ can be expressed by đđđđđđ (đđ đđđđđ đ đđđđ, so (32) are linear equations of đđđđđđ (đđ đđđđđ đ đđđđ, but they are similar to each other. However, every equation of (32) is independent with the (đđ đđđ linear equations obtained by the roots of |đ´đ´đđ (đ§đ§đ§đ§ in the interval (0, 1). îĸen all independent linear equations of đđđđđđ (đđ đđđđđ đ đđđđ are as follows: ķļĄķļĄđ´đ´0 ķļĄķļĄđ§đ§đđ ķļąķļąķļĄķļĄ = 0, đđ đđđđđđđđđđđđ đˇđˇ (1) đēđē0 (1) = đˇđˇ0 (1) . (33) đđ đđđđđđđđđđđđ đˇđˇ (1) đēđē0 (1) = đˇđˇ0 (1) . (34) îĸe steady-state probabilities of đđđđđđ (đđ đđđđđđđ đ đđđđ are obtained by solving (34). Using đđđđđđ (đđ đđđđđđđ đ đđđđ and (2), we obtain the other steady-state probabilities of đđđđđđđ (đđ đđđđđđđ đ đđđđ đđ đđđđđ đđ. 5.3. Mean Queuing Length. Aîer getting the probabilities đđđđđđđ (0 ≤ đđđđđđđđđđđđđđđđ in (19) we obtain đēđēđđ (1) (đđ đđđđđđđ đ đđđđ by solving (19) and obtain the steady-state availability (đ´đ´đ´ of the model by (20). From (31), we obtain đˇđˇ (đ§đ§) đēđēđđ (đ§đ§) = đđ , đˇđˇ (đ§đ§) đˇđˇ (đ§đ§) ≠ 0, |đ§đ§| ≤ 1, đđ đđđđđđđ đ đđđđ (35) îĸe PGF of đēđēđēđēđēđē is obtained by (7). Using the property of PGF [13] we obtain the steady-state mean queuing length is as follows: đŋđŋ đŋ đđđđ (đ§đ§) ķļĨķļĨ . đđđđ đ§đ§đ§đ§ đđ0 (đ§đ§) = ķļĄķļĄđđđđđķļąķļą đ§đ§ đ§đ§đ§đ§đ§2 , đđ1 (đ§đ§) = ķļĄķļĄđđđđđ2 +đđđđđķļąķļą đ§đ§ đ§đ§đ§đ§đ§2 − đđđ đđ2 (đ§đ§) = ķļĄķļĄ2đđđđđđ2 +đđķļąķļą đ§đ§ đ§đ§đ§đ§đ§2 − 2đđđ đđ0 (đ§đ§) = ķļĄķļĄđđ1 − đđ2 ķļąķļą đđ1,0 đ§đ§đ§ đđ1 (đ§đ§) = ķļĄķļĄđđđđđ2 − đđ1 ķļąķļą đđ1,0 đ§đ§ 2 + ķĩ ķĩ đđ ķļĄķļĄđđ1 − đđ2 ķļąķļą đđ2,2−đđ đ§đ§3−đđ − đđđđ1,0 đđđđ = đđđđ1,0 (đ§đ§ đ§đ§) + đ§đ§ ķļĄķļĄđđ1 − đđ2 ķļąķļą ķļąķļąđđ2,1 đ§đ§ đ§đ§đ§đ§2,0 − đđ1,0 ķļąķļą , 2 đđ2 (đ§đ§) = ķĩ ķĩ đđ ķļĄķļĄđđđđđ2 − đđ1 ķļąķļą đđ2,2−đđ đ§đ§3−đđ đđđđ 2 − ķĩ ķĩ đđđđđđ2,2−đđ đ§đ§2−đđ , đđđđ Further, from (29), (33) is equivalent to the follows: đˇđˇ0 ķļĄķļĄđ§đ§đđ ķļąķļą = 0, We analyze the case of đđ đđ and đđ đđ in this section. According to the above discussion, the determinant |đ´đ´đ´đ´đ´đ´đ´ (or đˇđˇđˇđˇđˇđˇ) of this case has only one root đ§đ§1 in the interval (0, 1). îĸe notations of this case are as follows: (36) đđ0 (đ§đ§) −đđ2 đ§đ§ đ§ đ´đ´ (đ§đ§) = ķļķļ −đđđđđđ1 (đ§đ§) −2đđ2 đ§đ§ķļ ķļ , −đđđđđđ2 (đ§đ§)ķļķļ ķļķļ 0 |đ´đ´ (đ§đ§)| = đ§đ§ (đ§đ§ đ§đ§) đđđđđđđđđđ −đđ2 0 −đđđđ ķļĄķļĄđđđđđ2 +đđđđđķļąķļą đ§đ§ đ§đ§đ§đ§đ§2 − đđđđđđ2 đ§đ§ ķļķļ , −đđđđ −đđđđđ đđ −đđđđđđđđķļķļ × ķļķļ ķļķļ đˇđˇ (đ§đ§) = ķļķļ ķļķļ đđđđđđđđđđ −đđ2 0 −đđđđ ķļĄķļĄđđđđđ2 +đđđđđķļąķļą đ§đ§ đ§đ§đ§đ§đ§2 − đđđđđđ2 đ§đ§ ķļķļ −đđđđ −đđđđđ đđ −đđđđđđđđķļķļ = − ķļĄķļĄđđđđđđđđđđķļąķļą ķļąķļąđ§đ§đ§đ§đ§đ§đ§đ§ķļąķļą ķļąķļą−đđđđđ ķļĄķļĄđđđđđđđđđđđđđķļąķļąķļąķļą + đ§đ§đ§đ§2 ķļĄķļĄđ§đ§đ§đ§ ķļĄķļĄ−2đđđđđđđđđđđđķļąķļą +2 ķļĄķļĄđđđđđđđđđđđđķļąķļą đđđđđđđđđđ2 ķļąķļą , (37) 8 Discrete Dynamics in Nature and Society then đđ đđđ2 0 đˇđˇ (1) = ķļķļ−đđ đđ đ đđ2 −2đđ2 ķļķļ ķļķļ−đđ đđđ đ đđ đđđ đđđđķļķļ ķļĄķļĄđđ đđđđđđ2 +đđķļąķļą đđ1,1 = đđđđ1,0 + đđđđ0,1 +đđđđ1,2 + ķļĄķļĄđđ1 +đđ2 ķļąķļą đđ2,1 , (38) ķļĄķļĄđđ đđđđđđ2 +đđķļąķļą đđ1,đđ = 2đđ ķļĸķļĸđđ2 đđ đ đđ2 ķļ˛ķļ˛ − đđ ķļĸķļĸ2đđ22 + 2đđ2 đđ đ đđ2 ķļ˛ķļ˛ , and đˇđˇđˇđˇđˇ đˇ đˇ is equivalent to or 2đđ ķļĸķļĸđđ2 đđ đ đđ2 ķļ˛ķļ˛ − đđ ķļĸķļĸ2đđ22 + 2đđ2 đđ đ đđ2 ķļ˛ķļ˛ > 0, 2đđđ ķļĄķļĄđđ2 + đđķļąķļą 2đđ đđ đ đđđ2 đđ < 2 2 = . 2 2 đđ 2đđ2 + 2đđ2 đđ đ đđ 1 + ķļĄķļĄđđ2 / ķļĄķļĄđđ2 + đđķļąķļąķļąķļą = đđđđ1,đđđđ + đđđđ0,đđ +đđđđ1,đđđđ + 2đđ2 đđ2,đđ , ķļĄķļĄđđ đđđđ1 ķļąķļą đđ2,0 = đđđđ1,0 +đđđđ2,1 , (39) 2 đđđđ1 (1) − 2đđ2 đēđē2 (1) = ķĩ ķĩ đđ ķļĄķļĄđđ1 −đđ2 ķļąķļą đđ2,2−đđ , 2 đđđđ ķļĄķļĄđđ đđđđ đđđđ2 ķļąķļą đđ2,đđ = đđđđ2,đđđđ + đđđđ1,đđ + 2đđđđ2,đđđđ , (40) đđ đđđ đđ đ đđ Using (42), we obtain ķļĄķļĄ2đđ1 + đđķļąķļą đđ2,0 − đđđđ1,0 . đđ đđ2,1 = (41) Equations (34) in this case are as follows: đˇđˇ0 ķļĄķļĄđ§đ§1 ķļąķļą = 0, đˇđˇ (1) đēđē0 (1) = đˇđˇ0 (1) . đđ đđđ đđ đđđ ķļĄķļĄđđ đđđķļąķļą đđ0,đđ = đđđđ0,đđđđ +đđ2 đđ1,đđ , đđ đđđ đđ đđđ ķļĄķļĄđđ1 −đđ2 ķļąķļą đđ1,0 −đđ2 ķļķļđđ ķļĄķļĄđđ2,1 đ§đ§ đ§đ§đ§đ§2,0 + đđ1,0 ķļąķļą đđ đđđđ đđđđ ķļķļ đđ đđđđ đđđđ đēđē (đ§đ§) = −đđđđ −đđđđ −đđđđ −đđđđ −đđđđđ đđ ķļĄķļĄđđ1 −đđ2 ķļąķļą đđ1,0 đđ1 (đ§đ§) đđ ķļĄķļĄđđ2,1 đ§đ§ đ§đ§đ§đ§2,0 + đđ1,0 ķļąķļą −đđ2 2 ķļĄķļĄđđđđđ2 + đđ đđđķļąķļą đ§đ§ đ§đ§đ§đ§đ§ −đđ đˇđˇ0 (đ§đ§) + đˇđˇ1 (đ§đ§) + đˇđˇ2 (đ§đ§) . đˇđˇ (đ§đ§) 0 ķļķļ ķļĄķļĄđđđđđ2 + đđ đđđķļąķļą đ§đ§ đ§đ§đ§đ§đ§ −đđđđđđ2 ķļķļ , đđ1 (đ§đ§) ķļķļ đˇđˇ1 (đ§đ§) = ķļķļ (45) Solving (45), we obtain đđ1,0 and đđ2,0 . Using (42), we obtain đđđđđđđ (đđ đđđđđđ, đđđđđđđ đ). Using (20) and (41), we obtain the steady-state availability đ´đ´. For the mean queuing lengths, we have đđ đđđ đđ đđđ ķļĄķļĄđđ đđđđđđ1 ķļąķļą đđ1,0 = đđđđ0,0 +đđđđ1,1 + 2đđ1 đđ2,0 , ķļķļ (43) (44) đˇđˇ (đ§đ§) = 0. Equations (2) in this case are as follows: ķļķļ đˇđˇ2 (đ§đ§) = ķļķļ (42) Using (41) and (43), đēđē0 (1), đēđē1 (1), and đēđē2 (1) are expressed in an algebraic expressions of đđ1,0 and đđ2,0 . If (40) is satisīŋŊed, we obtain đ§đ§1 by solving đđđđ ķļķļ đˇđˇ0 (đ§đ§) = ķļķļ đđ đđđ đđ đđđ đđ đđđ đđ đđđ ķĩ ķĩ đēđēđđ (1) = 1. ķļĄķļĄđđ đđđķļąķļą đđ0,0 = đđ1 đđ1,0 , đđ đđđ đđ đđđ ķļĄķļĄđđ đđđ1 +đđ2 +đđķļąķļą đđ2,1 = đđđđ2,0 + đđđđ1,1 + 2đđđđ2,2 , îĸe leî of (40) is the mean service quantities that all customers need per unit time. îĸe right of (40) is the mean service quantities that the two servers provide per unit time. So (40) is the necessary and suīŦcient condition of recurrence of the system. Equations (19) in this case are as follows: đđđđ0 (1) −đđ2 đēđē1 (1) = ķļĄķļĄđđ1 −đđ2 ķļąķļą đđ1,0 , đđ đđđ đđ đđđ −đđđđđ đđ (47) 2 0 đđđđđđđđķļķļ −2đđ2 ķļķļķļķļ , đđđđđđđđķļķļ ķļĄķļĄđđ1 −đđ2 ķļąķļą đđ1,0 đđ1 (đ§đ§) (46) ķļķļ ķļķļ , đđ ķļĄķļĄđđ2,1 đ§đ§ đ§đ§đ§đ§2,0 + đđ1,0 ķļąķļąķļķļ Using (36), we obtain the mean queuing lengths đŋđŋ is as follows: Discrete Dynamics in Nature and Society 9 TīĄīĸīŦīĨ 1: îĸe availability đ´đ´ and mean queuing length đŋđŋ (đđ đ đ, đđ đ đ, and đđ đđ). đđ1 = 0, đđ2 = 0.5, and đđ đđđđ L A đđ 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 đđ1 = 0.3, đđ2 = 0.5, and đđ đđ L A 3.0797 0.8951 2.3810 0.9058 1.9425 0.9147 1.6411 0.9224 1.4207 0.9290 1.2524 0.9348 1.1195 0.9399 1.0119 0.9443 đđ1 = 0.5, đđ2 = 0.5, and đđ đđđđ L A đđ 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 9.4410 6.1455 4.6345 3.7636 3.1952 2.7937 2.4944 2.2389 đŋđŋ đŋ = đđđđđđđđ ķļĨķļĨ đđđđ đ§đ§đ§đ§ 4.6956 3.5018 2.8136 2.3641 2.0464 1.8095 1.6256 1.4784 đđ1 = 0.5, đđ2 =1, and L 0.7423 0.7423 0.7423 0.7423 0.7423 0.7423 0.7423 0.7423 9.9094 5.7928 4.1374 3.2405 2.6761 2.2873 2.0025 1.7846 0.8209 0.8266 0.8315 0.8354 0.8383 0.8429 0.8458 0.8484 đđ đđđđ A 0.7412 0.7509 0.7608 0.7687 0.7756 0.7817 0.7872 0.7920 đđ1 = 0.5, đđ2 = 0.5, and đđ đđ L A 4.8505 3.6549 2.9650 2.5138 2.1947 1.9563 1.7710 1.6225 đđ1 =1, đđ2 = 0.8, and L 5.0503 3.7017 2.9584 2.4853 2.1565 1.9140 1.7273 1.5788 îĸe roots of đˇđˇđˇđˇđˇđˇđˇđˇ are as follows: 1 ķļĄķļĄđđđđ2 − 2đđđđ2 + 2đđ2 ķļĄķļĄđđđđđ đđđđđđđđđ2 ķļąķļąķļąķļą đ§đ§1 = 0.349123, × ķļķļ− ķļĸķļĸđđ ķļĸķļĸ−2đđđđđđđđ2 + 2đđđđđđđđđđđđđđ2 ķļ˛ķļ˛ +đđ2 ķļĸķļĸ−4đđđđđđđđ + 2đđđđđđđđđđđđđđđđ2 ķļ˛ķļ˛ķļ˛ķļ˛ × ķļĸķļĸđđ ķļĸķļĸđđ2 + 2đđ1 đđ đđđđ1 đđ2 ķļ˛ķļ˛ đđ1,0 +đđ ķļĸķļĸđđ2 + đđđđ2 + đđđđ1 + 2đđ1 đđ2 ķļ˛ķļ˛ ķļ˛ķļ˛2đđ2,0 + đđ2,1 ķļąķļąķļ˛ķļ˛ + đđ ķļĸķļĸ−đđ2 đđ đđđđđđ2 − 2đđ2 ķļĄķļĄđđđđđ đđđđđđđđđ2 ķļąķļąķļ˛ķļ˛ × ķļĸķļĸ ķļĸķļĸđđ2 − 2đđđđđđđđđđđđđđđđ2 + 2đđ1 đđ đđ2,0 = 0.297974. (51) đđ0,0 = 0.037844, đđ2,1 = 0.146599. (52) Using (42) we obtain Solving (41), we obtain đēđē2 (1) = 0.363647. + ķļĸķļĸđđ2 + đđđđ2 + đđđđ1 + 2đđ1 đđ2 ķļ˛ķļ˛ đđ2,1 đēđē1 (1) = 0.35602, (53) Finally, the availability and mean queuing lengths of this example are as follows: + ķļĸķļĸđđ2 − 2đđđđđ đđđđđ đđđđ2 − 2đđđđ2 + đđđđ1 (48) Numerical Example. Letting đđ đ đ, đđ đ đ, đđ đđ, đđ1 = 0.5, đđ2 =1, đđ đđ, and đđ đđ, we have 2đđđ ķļĄķļĄđđ2 + đđķļąķļą đđ 1 4 = < = . đđ 2 5 1 + ķļĄķļĄđđ2 / ķļĄķļĄđđ2 + đđķļąķļąķļąķļą2 đđ1,0 = 0.151375, đēđē0 (1) = 0.280333, −đđđđ1 + 2đđđđ1 + 2đđ1 đđ2 ķļąķļą đđ1,0 −đđđđ1 + 2đđ1 đđ2 ķļąķļą ķļąķļą2đđ2,0 + đđ2,1 ķļąķļą ķļąķļąķļąķļą . (50) and only đ§đ§1 in the interval (0, 1). Solving (45), we obtain 2 0.7734 0.7700 0.7671 0.7646 0.7623 0.7603 0.7585 0.7569 đ§đ§4 =8.33679, đ§đ§3 = 4.46896, 2 đ§đ§2 =1.84513, 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 đđ đđđđ A (49) đ´đ´ đ´đ´đ´đ´đ´đ´đ´đ´đ´đ´đ´đ´đ´đ´đ´đ´đ´đ´đ´đ´đ´ (54) îĸe other numerical results are shown in Table 1. All the system parameters in Table 1 satisfy (40). īŋŊe īŋŊnd that the mean queuing length (đŋđŋ) decreases with the increasing of the parameter đđ in Table 1, it is because of the greater service rate the less customers in the system. Furthermore, we īŋŊnd that the availability (đ´đ´) increases with the increasing of the parameter đđ, where đđ1 < đđ2 (the cases: 10 đđ1 = 0, đđ2 = 0.5; đđ1 = 0.3, đđ2 = 0.5; đđ1 = 0.5, đđ2 = 1); on the contrary, the availability decreases with the increasing of the parameter đđ, where đđ1 > đđ2 (the case: đđ1 = 1, đđ2 = 0.8); otherwise, the availability is constant, where đđ1 = đđ2 (the case: đđ1 = 0.5, đđ2 = 0.5). 7. Conclusions In Section 5.1, the inequality đˇđˇđˇđˇđˇđˇđˇ of îĸeorem 1 is the necessary and suīŦcient condition for the system to be positive recurrent, and a probability explanation of this condition is given by (40). We īŋŊnd that the idle time breakdown rate đđ1 does not appear in (40). îĸis is because the busy time breakdown rate đđ2 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 signiīŋŊcant eīŋŊect on the steady-state availability and steady-state queue length of the system. Acknowledgments îĸis paper is supported by the National Natural Science Foundation of China no. 71071133 and the Natural Science Foundation of Hebei Province no. G2012203136. References [1] A. Federgruen and K. C. So, “Optimal maintenance policies for single-server queueing systems subject to breakdowns,” Operations Research, vol. 38, no. 2, pp. 330–343, 1990. [2] K. H. 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