Discrete-time Geo/Geo/1 Queue with Negative Customers and Working Breakdowns

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1 Discrete-time GeoGeo1 Queue with Negative Customers and Working Breakdowns Tao Li and Liyuan Zhang Abstract This aer considers a discrete-time GeoGeo1 queue with server breakdowns and reairs. If the server is busy in a normal state, a breakdown is reresented by the arrival of a negative customer, and the failed server still works at a lower service rate rather than stoing the service comletely. Alying the matrix-analytic method, we derive the necessary and sufficient condition for the system to be stable. Using the robability generating function, we deal with the joint distribution of the server state and the number of customers in the system. The relation between our discrete-time model and its continuous-time counterart is also investigated. Finally, some numerical examles are resented to show the effect of various system arameters on the queueing characteristics. Furthermore, an oerating cost function is formulated and the otimum service rate in the working breakdown eriod is obtained. Index Terms GeoGeo1, negative customer, working breakdown, reair. I. INTRODUCTION Queues with negative arrivals, called G-queues, were first introduced by Gelenbe [1]. A negative customer will vanish if it arrives to an emty queue, and negative customers cannot accumulate in a queue and do not receive services. In some cases, an arrival negative customer makes the server break down when the system is in a normal state. Queues with server breakdowns can be alied in manufacturing systems, roduction systems, telecommunication systems, inventory systems and comuter systems. Wang and Zhang [] investigated a GeoG1 retrial queue with negative customers, the server is subject to failure due to the negative arrivals. Wu and Lian [3] analyzed an MG1 retrial G-queue with unreliable server under Bernoulli vacation schedule. Using the matrix geometric method, Rakhee et al. [4] studied a GeoGeo1 queue with unreliable server, the breakdown of the server is reresented by the arrival of a negative customer. For more queueing models with server breakdowns and reairs, readers can refer to Do [5], Yang et al. [6], Gao and Wang [7] and Tsai et al. [8]. In the study of queueing systems, it is usually assumed that the server stos the service comletely in a breakdown eriod. However, in many ractical cases, the failed server still can serve a customer at a lower service rate. This tye of breakdown is called as working breakdown introduced by Kalidass and Kasturi [9], and the MM1 queue they analyzed can be alied to studying the behavior of communication system or machine relacement roblem. Liu and Song [1] Manuscrit received January 4, 17; revised Setember 4, 17. This work was suorted by the National Natural Science Foundation of China T. Li is with the School of Mathematics and Statistics, Shandong University of Technology, Zibo, 5549, China liltaot@16.com. L. Zhang is with the Business School, Shandong University of Technology. extended the model in [9] to an M X M1 queue, the customers arrive in batches. Kim and Lee [11] discussed an MG1 queue with disasters and working breakdowns, all resent customers leave the system when a disaster haens. Recently, Jiang and Liu [1] investigated a GIM1 queue with disasters and working breakdowns in a multi-hase service environment. Using the matrix geometric method, Ma et al. [13] comuted the steady-state distribution of an MM1 queue with multile vacations and working breakdowns. Readers can also refer to Liou [14], [15] and Yen et al. [16]. Note that working breakdown is different from working vacation introduced by Servi and Finn [17]. A working vacation is taken only when the system becomes emty, while a working breakdown can occur at any time oint. Some authors like Zhang and Liu [18] and Rajadurai et al. [19], [] have studied working vacation queues with unreliable server, the server is subject to breakdown due to the negative arrivals. The concet of working breakdown introduced by Kalidass and Kasturi [9] does make sense in real life. For examle, when a comuter is infected by a virus, it may still be able to erform but in a slower service rate. Parallel to continuoustime queues, discrete-time models are more suitable to analyze comuter systems or communication systems, the reason is that these systems oerate on a discrete-time basis the events arrival of ackets and their forward transmissions only take lace at regularly saced eochs. A detailed discussion and alication of discrete-time queues can be found in Woodward [1] and Ramasamy et al. []. U to now, a few authors have discussed continuous-time queues with working breakdowns, but their discrete-time counterarts seem to receive very little attention in the literature. Insired by the natural and reasonable alications of discrete-time queues, we deal with a GeoGeo1 queue with working breakdowns in this aer, a breakdown occurs due to a negative arrival. In order to make a comarison with the continuoustime system [9], the killing disciline is not considered in this model. This aer is organized as follows. Section gives a brief descrition of the model. Using the matrix-analytic method, the stable condition is obtained. In Section 3, we deal with the steady state joint distribution of the server state and the number of customers in the system. Section 4 gives the relation between our discrete-time queue and its corresonding continuous-time model. In Section 5, some numerical examles and cost otimization analysis are resented. Finally, Section 6 concludes the aer. II. MODEL DESCRIPTION AND STABILITY CONDITION In this aer, for any real number x [, 1], we denote x = 1 x. We consider a early arrival system, and the Advance online ublication: 17 November 17

2 GeoGeo1 queue with two tyes of customers and working breakdowns is given as follows: 1 Assume a otential dearture due to the comletion of a service occurs in n, n, and the distribution of service time S b in a normal eriod is P {S b = k} = µ µ k 1, k 1, < µ < 1. A otential ositive customer arrival takes lace in n, n +, and inter-arrival times of ositive customers follow the geometric distribution: P {A = k} = k 1, k 1, < < 1. There are at least two ossible choices for the location of a negative arrival: i in n, n + and immediately after a otential ositive arrival; ii in n, n + and immediately before a otential ositive arrival. In this aer, we choose i as Atencia and Moreno [3] did. Inter-arrival times of negative customers follow the geometric distribution: P {B = k} = δ δ k 1, k 1, < δ < 1. 3 In a normal eriod, the arrival of a negative customer makes the server break down if the server is busy i.e., there are customers in the system at that moment. When the server breaks down, the service rate decreases, and the distribution of service time S w in a breakdown eriod is P {S w = k} = k 1, k 1, < < 1, < µ. 4 When the server breaks down, a reair rocedure starts immediately. We assume that the beginning and ending of reair occur at the slot n, and the reair times follow the geometric distribution: P {R = k} = θ θ k 1, k 1, < θ < 1. 5 The arriving negative customer has nothing to do with the server when the server is free or under reairing, and negative customers cannot accumulate in a queue and do not receive services. We assume that inter-arrival times, service times and reair times are mutually indeendent. Here we give a ractical alication of this model. In a comuter system, date ackets arrive at the system according to a Bernoulli rocess with arameter. When the comuter system is oerating in a normal state, the rocessing time service time for each data acket is geometrically distributed with arameter µ. The comuter system may be subject to the invasion of a virus during the normal oeration eriod, and the time interval until the resence of virus follows a geometric distribution with arameter δ. If the comuter system is invaded by a virus, the CPU of the comuter will not sto running comletely and still can work at a lower seed. Under this situation, the rocessing time for each data acket is governed by a geometric distribution with arameter < µ. Meanwhile, the antivirus software begins to reair the system until the virus is cleared, and the reair times follow a geometric distribution with arameter θ. Let J n be the state of server at time n +, and Q n be the number of customers in the system at time n +. There are two ossible states of the single server as follows: i J n = 1, the server is in a normal eriod at time n + ; ii J n =, the server is defective in a working breakdown eriod at time n +. Then, {J n, Q n } is a Markov chain with state sace Ω = {j, k, j = 1,, k }. Remark 1: In order to make a comarison with the continuous-time model Kalidass and Kasturi [9], we do not consider killing disciline in this aer. If we consider removal disciline, such as RCH, i.e., the negative arrival can remove a customer being in service, the system will have a similar solution but is not ursued in the resent work. Using the lexicograhical sequence for the states, the transition robability matrix can be written as P = A A 1 B 1 A 1 A A A 1 A , µ δ µδ A = ; A θ θ = ; θ δ θ + θδ µ δ δ B 1 = ; A θ θ 1 = ; θ δ θ + θδ µ δ + µ δ µ δ + µδ A 1 = ; θ δ + θ δ θ + θ + θ δ + θδ µ δ µ δ A =. θ δ θ + θ δ Due to the block structure of transition robability matrix, {J n, Q n } is called a QBD rocess. Theorem 1: The QBD rocess {J n, Q n } is ositive recurrent if and only if θ δµ µ > δ. Proof: Let δ δ A = A + A 1 + A = θ δ θ + θδ, the Theorem 7..3 in [4] states that the QBD is ositive recurrent if and only if πa e > πa e, e is a column vector with all elements equal to one, and π is the unique solution of the system πa = π, πe = 1. After some algebraic maniulation, we have π = θ δ θ δ+δ, δ θ δ+δ, and the QBD rocess is ositive recurrent if and only if µθ δ + δ > µθ δ + δ θ δµ µ > δ. III. STEADY STATE ANALYSIS If θ δµ µ > δ, let J, Q be the stationary limit of the rocess J n, Q n, and denote P j,k = P {J = j, Q = k} = lim n P {J n = j, Q n = k}, j, k Ω. Using the transition robability matrix P, the balance Advance online ublication: 17 November 17

3 equations governing the system are given by = + µ P 1,1 + θ P, + θ P,1, 1 P 1,1 = δ + µ δ + µ δp 1,1 + µ δp 1, + θ δp, + θ δ + θ δp,1 + θ δp,, P 1,k = µ δp 1,k 1 + µ δ + µ δp 1,k + µ δp 1,k+1 + θ δp,k 1 + θ δ + θ δp,k + θ δp,k+1, k, 3 P, = θ P, + θ P,1, 4 P,1 = δ + µ δ + µδp 1,1 + µ δp 1, + θ + θδp, + θ + θ + θ δ + θδp,1 + θ + θ δp,, 5 P,k = µδp 1,k 1 + µ δ + µδp 1,k + µ δp 1,k+1 + θ + θδp,k 1 + θ + θ + θ δ + θδp,k + θ + θ δp,k+1, k. 6 Define artial robability generating functions P 1 z = k=1 P 1,kz k, P z = k= P,kz k. Multilying and 3 by aroriate owers of z, and summing over k 1, we can obtain µ δz + 1 µ δ µ δz µ δ P 1 z θ δ z + + z + P z = δzz 1 + θ δz + z 1P,. 7 Multilying 4, 5 and 6 by aroriate owers of z, and summing over k, we can get µδz + µ δ + µδz + µ δ P 1 z + θz + 1 θ θz θ P z θδ z + + z + P z = δzz 1 + θ + θδz + z 1P,. 8 Solving Eqs. 7 and 8, we have [ P 1 z = δz θz + θ zz 1 + θ δzz + P, ]ϕz, 9 [ P z = δz + θ δµ µzz 1 + δz z + P, ]ϕz, 1 ϕz = δµ µz zz 1 + δz z + θ δµ µz z + z +. Lemma 1: If θ δµ µ > δ, the equation ϕz = has a unique root z = r 1 in the interval,1. Proof: We can easily obtain ϕ = µ θ δ <, ϕ1 = δ + θ δµ µ >, µ ϕ = µ δ µ µ µ <, ϕ+ = + the coefficient of z 3 is µ θ δ >. Thus, the three roots of ϕz lie in, 1, 1, µ µ and µ µ, +, and ϕz = has a unique root z = r 1 in the interval,1. From Lemma 1, the numerator of P 1 z must vanish at z = r 1, we have δr 1 θr 1 + θ r 1 r 1 1 which means + θ δr 1 r 1 + P, =, P, = θ r 1 1 r 1 θr 1 θr 1 + = Lr Therefore, using 11, we have the following theorem. Theorem : [ P 1 z = δz θz + θ zz 1 + θ δzz + Lr 1 ] ϕz, [ P z = δz + θ δµ µzz 1 + δz z + Lr 1 ] ϕz, is determined by the normalization condition which leads to = + P P 1 = 1, ϕ1 δ + θ δ + Lr 1 + ϕ1. 1 The robability generating function of the system size is given by P z = + P 1 z + P z [ = ϕz + z δ + θ δz + θ δ zz 1 + δ + θ δz + θ δµ µzz 1 z + Lr 1 ] ϕz. 13 Let W denote the sojourn time of a customer in the system, and W s is the Lalace Steiljes transform of W. Then, W and Q have the following classical relationshi see [5] P z = W z +. Using 13, we can get [ W s = ϕ s + s δ + θ δs + θ δ ss 1 + s δ + θ δs + θ δ [ µss 1 Lr 1 ] ϕ s ], s ϕ = δ µs ss 1 + δs s + θ δs µs s +. The robability that the server is busy in the normal state is given by P N = P 1 1 = θ δ + θ δlr 1. ϕ1 Advance online ublication: 17 November 17

4 The robability that the server is busy in the defective state working breakdown eriod is given by P W = P 1 P, = δ + δ ϕ1lr 1. ϕ1 The robability that the server is free in the normal state and in the defective state are given by and P,, resectively. Let E[L j ] denote the average number of customers in the system when the server s state is j, j = 1,. From Theorem, after some calculations we can get E[L 1 ] = lim z 1 P 1z = E[L ] = lim z 1 P z = [ θ δ + Lr 1 + θ δ1 + Lr 1 + θ δ ϕ1 θ δ + Lr 1 ϕ 1 ] ϕ 1, [ δ + Lr 1 + δ1 + Lr 1 + θ δµ µlr 1 ϕ1 δ + Lr 1 ϕ 1 ] ϕ 1, ϕ 1 = δµ µ + δ + θ δ µ µ + µ. The exected number of customers in the system is given by = lim z 1 P z = E[L 1 ] + E[L ]. Let E[W ] be the exected sojourn time of a customer in the system, using Little s formula, E[W ] =. IV. RELATION TO THE CONTINUOUS-TIME QUEUEING SYSTEM This section discusses the relation between our discretetime queue and its corresonding continuous-time system [9]. If it is assumed that the time is slotted into sufficiently small intervals of equal length ϵ, the continuous-time queue [9] can be aroximated by our discrete-time queueing system for which = λϵ, δ = αϵ, µ = µϵ, = µ 1 ε, θ = βϵ. From Eqs. 9 and 1, we can obtain P 1 z = lim P 1 z ϵ = [ λz βz + µ1 λzz 1 P1, + µ 1 βz P, ] ϕz, P z = lim P z ϵ = [ λ αz P1, + µ 1 µ λzz 1 + αz P, ] ϕz, ϕz = µ λz µ 1 λzz 1 + αz µ 1 λz + βz µ λz. From Eqs. 11 and 1, we can have P, = lim P, = λ µ 1 λ r 1 1 r 1 β λ r 1 ϵ µ 1 β = L r 1, ϕ1 = lim = ϵ α + β λ + µ 1 L r 1 + ϕ1, =.4 = δ Fig. 1. The exected queue length versus δ =.4 = δ Fig.. The robability of server being free in the normal state versus δ. ϕ1 = α µ 1 λ + β µ λ and r 1 is the unique root of the equation ϕz = in the interval,1. After some calculations, we can find the exressions of P1 z, P z, P, and are agreement with the exressions obtained in Kalidass and Kasturi [9]. V. NUMERICAL RESULTS In this section, we resent some numerical examles to illustrate the effect of varying arameters on some crucial erformance measures of our model. Moreover, a cost minimization roblem is also considered. Under the stable condition, all the comutations are done by develoing rogram in Matlab software and the values of arameters are arbitrarily chosen as µ=.8, =.4, θ=.3, =.5 and δ=., unless they are considered as variables in the resective figures. A. Sensitivity analysis The effect of δ on the exected queue length and the robability of server being free in the normal state are resented in Fig.1 and Fig., resectively. We can find that increases with increasing values of δ, while decreases as δ increases. The reason is that as δ increases, the system is more likely to break down and the service rate will decrease, which in turn increases the system queue length. Moreover, the effect of δ on is not obvious when the Advance online ublication: 17 November 17

5 .5 =.4 = θ=.3 θ=.5 θ= Fig. 3. The exected queue length versus Fig. 5. The exected queue length versus =.4 = θ=.3 θ=.5 θ= P,.45 P W Fig. 4. The robability of server being free in the defective state versus. Fig The robability of server being busy in the defective state versus value of is large. An esecial case is δ, i.e., the server can not break down, and the system will be in normal state with robability 1, we can see that has no effect on and. Fig.3 illustrates that increases as increases, which agrees with the intuitive exectation. When is small, which means the number of customers in the system is small, it can be found that the effect of on is not obvious. In Fig.4, we see that as increases, the robability of server being free in the defective state P, first increases and then decreases. However, the effect of on P, is not obvious. For examle, if we choose =.4, with the change of, the values of P, only varies from.3 to.47. The main reason is that the failure rate can be regarded as δ=., the mean reair time is 1θ, and the server can not break down if the system is emty. Fig.5 and Fig.6 rovide the effect of on the exected queue length and the robability of server being busy in the defective state P W, resectively. As exected, and P W both decrease with increasing values of. The effect of is more obvious when θ is smaller, this is due to the fact that the exected reair time is 1θ, and the defective system will be reaired in a longer time. In Fig.5, an esecial case is µ, i.e., the lower service rate equals to the normal service rate, it can be observed that θ has no effect on. As shown in Fig.7, since < µ, it is obvious that decreases as θ increases, and the effect of θ on is not obvious when θ is large. The reason is that the mean reair time is becoming shorter with the increasing reair rate θ, and therefore the waiting customers have a greater chance to be served by normal service rete, which can reduce the µ=.7 µ=.8 µ= θ Fig. 7. The exected queue length versus θ. Advance online ublication: 17 November 17

6 system queue length. Fig.8 indicates that the robability of server being busy in the normal state P N increases with the increment of θ, this is also because that the duration of server in the defective state is shorter for larger value of θ. Further, as intuitively exected, for a fixed reair rate θ, and P N both decrease as µ increases. B. Cost analysis In ractice, from the ersective of economic rofit, queueing managers are always interested in minimizing oerating cost of unit time. Therefore, in this subsection, we establish a cost function to search for the otimal service rate, so as to minimize the exected oerating cost function er unit time. Define the following cost elements: C L =cost er unit time for each customer resent in the system; C µ =cost er customer served by the normal service rate µ; C =cost er customer served by the lower service rate ; C θ =fixed cost er unit time when the server is in a reair working breakdown eriod. Based on the definitions of each cost element listed above, the exected oerating cost function er unit time is given by min : f = C L + C µ µ + C + C θ θ. 14 Because the exected oerating cost function er unit time is highly non-linear and comlex, we can use the arabolic method to find the otimum value of, say. The essence of the arabolic method is to generate a quadratic function through the evaluated oints in each iteration, and the objective function fx is aroximated by the quadratic function in generating an estimate of the otimum value. According to the olynomial aroximation theory, the unique otimum of the quadratic function agreeing with fx at 3-oint attern {x, x 1, x } occurs at x = 1 fx x 1 x + fx 1 x x + fx x x 1 fx x 1 x + fx 1 x x + fx x x 1. The stes of the arabolic method are given as follows [6]: P N µ=.7 µ=.8 µ=.9 Ste 1: Choose a starting 3-oint attern {x, x 1, x } along with a stoing tolerance ε, and initialize the iteration counter i=. Ste : Comute x according to the above equation. If x x 1 ε, sto and reort aroximate otimum solution x. Ste 3: If x x 1, go to Ste 4. If x>x 1, go to Ste 5. Ste 4: If fx 1 is less than f x, udate x x. Otherwise, relace x 1 x, x x 1. Either way, advance i = i + 1, and return to Ste. Ste 5: If fx 1 is less than f x, udate x x. Otherwise, relace x 1 x, x x 1. Either way, advance i = i + 1, and return to Ste. Exected oerating cost er unit time Fig The effect of on the exected oerating cost er unit time. Assume C L =3, C µ =18, C =1 and C θ =5, Fig.9 shows that there is an otimal service rate to make the cost minimize. Using the arabolic method and the error is controlled by ε=1 5. After three iterations, Table 1 shows that the minimum exected oerating cost er unit time converges to the solution =.5686 with a value f = VI. CONCLUSION This aer generalizes the model of Kalidass and Kasturi [9] to a GeoGeo1 queue. During the breakdown eriod, the service still continues at a lower rate. Using the matrixanalytic method, we obtain the condition of stability. The robability generating function of the number of customers in the system is also discussed. Moreover, various system erformance measures are develoed, and the effect of some arameters are examined numerically. The novelty of this investigation is the first time to consider working breakdowns in a discrete-time queue. For future study, one can analyze a similar system with retrial customers or extend this model to a GeoG1 queue θ Fig. 8. The robability of server being busy in the normal state versus θ. REFERENCES [1] E. Gelenbe, Random neural networks with negative and ositive signals and roduct form solution, Neural Comutation, vol. 1,. 5-51, [] J. Wang and P. Zhang, A discrete-time retrial queue with negative customers and unreliable server, Comuters & Industrial Engineering, vol. 56, , 9. [3] J. Wu and Z. Lian, A single-server retrial G-queue with riority and unreliable server under Bernoulli vacation schedule, Comuters & Industrial Engineering, vol. 64, , 13. Advance online ublication: 17 November 17

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