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1 University of Wurzburg Institute of Computer Science Research Report Series Sojourn Time Distribution of the Asymmetric M=M=1==N { System with LCFS-PR Service M. Mittler and R. Muller Report No. 94 November 1994 Lehrstuhl fur Informatik III, Universitat Wurzburg, Am Hubland, D { Wurzburg Tel.: , Fax: mittler@informatik.uni-wuerzburg.de Abstract: In this paper we present a recursive denition of the sojourn time probability distribution function of the M=M=1==N{LCFS-PR system. Applications of this model can be found in telecommunication systems and in manufacturing systems. By means of a transformation method we obtain the rst two moments of sojourn times in a symmetric system with an arbitrary number of messages. This result is generalized regarding heterogeneous messages with dierent service requirements yet it is applied to only two messages. Finally we numerically compare the variance of sojourn times to the FCFS case for which we derive an explicit expression. Keywords: queueing theory, M=M=1==N, LCFS-PR, FCFS, sojourn times, variance, machine repairman model

2 1. Introduction The M=G=1 system with Last-Come-First-Served{Preemptive-Resume (LCFS-PR) service discipline has been subject of many investigations due to its outstanding property as a product form queueing node (see BCMP [2]). Applications of the M=G=1{LCFS-PR system can be found in the performance analysis of computer systems as stated by Sauer and Chandy [16]. They report on the state transition diagram of the M=M=1 queue with LCFS-PR service. Durr [8] gives an explicit formula for the waiting time distribution of the M=G=1 system with an arbitrary number of priorities having either LCFS-PR or LCFS nonpreemptive priority queueing discipline. A comparison of the rst two moments of the waiting time for any class of priorities in an M=G=1 queue where customers are served in LCFS-PR manner among the same priority classes can be found in Takagi [20]. He also presented a vacation system approach to this conguration. In contrast to the aforementioned open system with an innite number of sources, there are many publications on the same system where messages leaving the system are immediately fed back to the input, thus yielding a closed network with a nite number of messages. The best known application of this model is the so-called Machine Interference Problem (MIP) where N machines are subject to failures. Upon failure the machines wait for being repaired by a repairman which is represented by a service station in the closed queueing network. In this model failure times correspond to service times at dedicated nodes whereas repair times are incorporated in the service times at the service facility. Therefore the resulting closed queueing network contains N customers where each customer belongs to a dierent class and follows a dierent route. With the repair strategies FCFS and LCFS-PR, resp., this queueing network has the product form. A further application can be found in telecommunication systems where a source needs a so-called thinking time to generate a new message (see Takagi [19]). Having been generated, a message is passed to the service facility where it is transmitted. In Kendall's notation the MIP reads as A=B=m==N where A denotes the failure time distribution, B the repair time distribution for a MIP with N machines, and m the number of repairmen. An alternative notation not used in this paper is A=B=m=N=N where the queueing capacity is given explicitly (see Allen [1]). A queue capacity of N ; 1 would also be sucient since one customer can be buered in the server. Next we shall review the literature considering the MIP. There is a considerable amount of publications on the MIP regarding the queueing discipline FCFS. See for example the review by Stecke and Aronson [17], chapters and of Allen [1], chapter 2 of Buzacott and Shantikumar [5], and chapter 4 of Takagi [21]. Takacz [18] presented the probability distribution function (pdf) of waiting times of the M=G=1==N system considering identical machines which are repaired in the order of their failures (FCFS). Further, he derived the mean waiting time. Kobayashi stated 1

3 the waiting time pdf of the M=M=m==N system which is given in the solutions manual to [12] Boyse and Warn [4] presented the mean waiting time and the mean queue length of the M=D=m==N system. Buzacott and Shantikumar [5] derived the steady state probabilities of the E k =M=1==N, M=E k =1==N, M=M=m==N, and ~M=M=m==N systems. Though not mentioned explicitly, the repair strategy in all models was FCFS. An advanced investigation has recently been published by Takagi [21]. He derived the waiting time pdf of the M=G=1==N system with either FCFS, LCFS, or Random Order of Service (ROS). Only the publications of Kelly [10], Buzacott and Shantikumar [5], and Takagi [21] deal with dissimilar failure times of the machines. To the authors' knowledge there is currently no paper on this model with heterogeneous service requirements. However, for machines with dissimilar failure times either only the stationary state probabilities (see [10] and [5]) or only the waiting time pdf is presented (see Takagi [21]), FCFS repair discipline provided. Further, Kelly [10] investigates generally distributed failure times. By the argument of quasi-reversibility, he shows that the stationary state probabilities are the same as for exponentially distributed service times. For the special case of the MIP we give an alternative proof of this statement later in this paper. The aim of this paper is as follows. Since the LCFS-PR queueing discipline is well known for the worst case variance of sojourn times in open single stage queueing systems (see [8]) we derive the sojourn time distribution for a closed queueing network which represents the MIP with LCFS-PR service assuming failure times and repair times to be exponentially distributed. In the simplest case of two machines we allow for both nonidentical failure and repair times. We also investigate the variance of sojourn times in the FCFS case and compare the results to the LCFS-PR queueing discipline. The motivation of this analysis can be found in MIPs where the repairman has to interrupt the repair of any machine if another machine has just broken down and immediately requires service. We will show that the LCFS-PR queueing discipline is worth investigating, because, as Heyman and Sobel [9] state, "it leads to simple (and often surprising) analytical solutions". The paper is organized as follows. In Sec. 2 we review the considered model. In order to clarify the basic idea behind the analysis, we start with a symmetric system consisting of only N = 2 machines in Sec A recursive denition of the sojourn time pdf for an arbitrary number of machines is given in Sec. 3.2 for identical machines as well as for the rst two moments of the time to repair. These results are extended in Sec. 3.3 for two heterogeneous machines with dierent repair time requirements. In Sec. 4 we derive a simple expression for the variance of sojourn times in the M=M=1==N{FCFS system. Numerical results considering the queueing disciplines FCFS and LCFS-PR are presented in Sec. 5. 2

4 2. The model The model considered in this paper is shown in Fig. 1. There are N customers in the system with each customer belonging to a dierent class. While passing the so-called dedicated node i the customer of class i represents the failure time of machine i with mean ;1 i. Note that there are no queues at the dedicated nodes 1 to N since there is only one customer per customer class in the network. The repair times are incorporated in the service times at the service facility. The mean service time for machine i is ;1 i. According to Martz and Waller [14] the waiting time in the repair station is referred 1 N ; 1 places i 2 N time to wait repair time time to fail TTW B TTF TTR time to repair Fig. 1: Queueing network with corresponding random variables to as the Time To Wait (TTW) and the failure time is referred to as the Time To Fail (TTF). Further, we use their denotation Time To Repair (TTR) for the sojourn time distribution of the service facility since the sojourn time at the repair node is just the time from the instant of failure until the instant of repair completion. The random variables are also given in Fig. 1. Note the dierence of the time to repair and the repair time. The latter is simply the service time at the repair station. The following theorem states an important relationship between the pdf of failure times and repair times on the one hand, and the stationary state probabilities on the other hand: 3

5 Theorem 1 The stationary state probabilities of the M=M=1==N system with LCFS- PR service are insensitive to the pdf of both failure and repair times except their mean. Proof : Since there are N dierent classes of customers in the network where each class contains only a single customer and the failure time of machine i corresponds to the service time at the dedicated node i, we are able to apply the service discipline LCFS- PR at the dedicated nodes without breaking the product form property of the network. Following de Souza e Silva and Muntz [7] the LCFS-PR queueing discipline has the socalled station balance property. The station balance property states that under certain conditions not reported here "for every station the probability that a customer enters that station must be proportional to the rate of service at that station" (see Chandy and Martin [6]). In this context, a station denotes a waiting place within the queue. It is clear that this condition implies that a customer entering the system must immediately receive service. Chandy and Martin have shown that the stationary state probabilities are insensitive to the service time pdf of the nodes having the station balance property except the mean service time. Thus, the proof is complete. An alternative proof of the insensitivity of the stationary state probabilities to the pdf of failure times except the mean failure time can be found in Kelly [10]. Under the assumption that failure times are generally distributed and repair times are exponentially distributed he showed that "the queue containing running machins is equivalent to one which would in isolation behave as an M=G=1 queue. Thus, the queue remains quasireversible" the product form equation for the stationary state probabilities remains unchanged compared to the case with exponentially distributed failure times. The quasireversibility and its application to queueing networks is also due to Kelly [10]. De Souza e Silva and Muntz [7] state that "informally, a quasi-reversible queue is one for which the past departures, the current state, and the future arrivals are mutually independent. A property that follows from this denition is that the arrival process and the departure process are Poisson". As a consequence, we are able to replace the generally distributed failure and repair times by exponentially distributed failure and repair times and still obtain the same stationary state probabilities. Of course, the pdf of the sojourn time depends on the pdf of both failure and repair times. We derive the sojourn time pdf of the M=M=1==N system with LCFS-PR service in the next section. However, the consideration of generally distributed failure and repair times is left to future research. 3. Recursive Denition of the Sojourn Time PDF The derivation of the sojourn time distribution of a particular customer is not as simple as one might expect since its sojourn time depends very strictly on the number of customers arriving while he passes the repair station. This is the case because a new 4

6 arriving customer preempts the service time of any other customer currently receiving service. In order to clarify the basic idea behind the analysis we start with the derivation of the sojourn time pdf of a symmetric system, i.e. a system with only two identical machines, i =, and identical service requirements, i =, i = 1 2. Here, we use the denotation of the machine repairman model such that customers are referred to as machines. 3.1 Symmetric System with N = 2 Machines Given a tagged machine, two dierent situations may arise upon the breakdown of the tagged machine: (1) The tagged machine nds the other machine present in the repair facility receiving service. (2) The repair facility is empty upon arrival of the tagged machine. In the rst case the tagged machine preempts service. After service time completion the tagged machine leaves the service station and the service of the second machine is resumed. In this case the sojourn time TTR of the tagged machine is simply the service time: TTR (0) (t) = B(t) (3.1a) where TTR (n) denotes the random variable for the sojourn time of a machine which nds N ; n ; 1 machines broken down in the repair facility upon its arrival, i.e. when further n machines are still working. In the second case the server immediately starts to repair the tagged machine. The service time maybeinterrupted with probability p = =(+)by the breakdown of the second machine. Then the service of the tagged machine will be resumed after the service time completion of the second machine. But even in this case the tagged machine may be preempted once again and the process continues. Since repair times are exponentially distributed the service time after service resumption is also exponentially distributed with the same mean. The number of service preemptions before the successful service attempt, K, is geometrically distributed with parameter p, i.e: P [K = k] = (1 ; p)p k, k 0. Consequently, in case (2) the sojourn time pdf of the tagged machine through the repair station meets the following equation: TTR (1) (t) = B(t) 1X k=0 P [K = k] B (k) (t) (3.1b) where B (k) (t) denotes the service time pdf B(t) convoluted k times with itself, and B (0) (t) 1. 5

7 Finally, we obtain the total sojourn time pdf by taking into account the probability q n that a newly arriving machine nds already n machines in the service facility. The sojourn time pdfattheservice facility in the case of N = 2 machines is TTR(t) = q 1 B(t) + q 0 B(t) = 1X n=0 1X k=0 P [K = k] B (k) (t) q 2;n;1 TTR (n) (t): (3.2) Clearly, following the arrival theorem of Lavenberg and Reiser [13], in this closed queueing network the stationary arrival probabilities q n correspond to the stationary state probabilities n with one customer removed from the network, i.e. q n (N) = n (N ; 1), 0 n N ; 1. To complete the picture we shall report the stationary state probabilities for a system with N machines (see [3] and [1]): n = n;n (N ; n)! NP ;k k! k=0 0 n N (3.3) where = =. Alternatively, these probabilities can be obtained by the product form equation given by BCMP [2]. Looking at that equation, it becomes clear that the stationary state probabilities for the LCFS-PR case are the same as for the FCFS case. 3.2 System with N Machines In this subsection we extend the results presented abovetoanetwork consisting of N > 2 identical machines and identical service requirements. Kendall's notation of the resulting system is M=M=1==N{LCFS-PR. Further, we derive the rst two moments of the total sojourn time pdf. The major changes arising out of the extension to N machines concern the probability that the service of any machine will be preempted. Given n working machines then the service given to any machine will be preempted with probability p n = n + n = 1 ; + n : (3.4) Also given n working machines, K (n) denotes the random variable for the number of service interruptions before the successful service attempt. Obviously, K (n) is geometrically distributed with parameter p n from eqn. (3.4). The repair of the preempting machine may again be interrupted by one of the remaining n ; 1working machines. This process may continue until all machines are broken down. Taking into consideration the sojourn time of a tagged machine and the sojourn time of all machines which preempt the tagged machine we may state the following 6

8 Theorem 2 The pdf of the total sojourn time, TTR(t), of the M=M=1==N{LCFS-PR system is TTR(t) = N;1 X n=0 with TTR (n) (t) = B(t) q N;n;1 TTR (n) (t) 1X k=0 P [K (n;1) = k] TTR (k) (n;1) (t): (3.5a) (3.5b) It is clear that, given n working machines, the sojourn time of the tagged machine just entering the repair station consists of its own repair time and the k-fold sojourn time of the next machine which breaks down and therefore preempts the service of the tagged machine. We denote this k-fold sojourn time pdf by TTR (k) (t). The recursive denition (n;1) of the pdf of sojourn times is presupposed by the sojourn time of a machine nding all other machines in the service station, i.e. n = 0(seeeqn. (3.1a)). As mentioned above the considered network has the product form. If we take into account the stationary state probabilities given by the product form equation in [2] the mean number of customers in the repair node does not change if we assume the repair station to be either of M=M=1-FCFS type or of M=G=1-LCFS-PR. Thus, for exponentially distributed repair times the mean waiting times are by Little's law the same for the FCFS and the LCFS-PR case. This statement contradicts Kleinrock's conservation law for the M=G=1 queue with priorities which states that for any work-conserving but nonpreemptive queueing discipline the sum of mean waiting times weighted by the relative trac intensity never changes (see [11]). A queueing discipline is called work-conserving if "(i) no servers are free when a customer is in the queue and (ii) the discipline does not aect the amount of service time given to a customer or the arrival time of any customer" (cf. Heyman and Sobel [9]). Notice that although the LCFS-PR service discipline is work-conserving but not nonpreemptive the mean sojourn times are the same as in the FCFS case. The reason therefore are the equations of the mean value analysis for the mean response time which are the same for M=G=1{LCFS-PR and M=M=m{FCFS queues (see Reiser and Lavenberg [15]). Following Kobayashi [12] and Allen [1], the mean sojourn time of the M=M=1==N- FCFS system is E[ TTR] = N (1 ; 0 ) ; 1 : (3.6) Finally, we derive the variance of sojourn times by taking the generating function of the distribution of the number of service preemptions (which is of course the geometric distribution) G(z) = 1 ; p 1 ; zp (3.7a) 7

9 and by taking the Laplace-Stieltjes-Transform (LST) of the service time pdf B (s) = + s : (3.7b) Setting z = B (s), then given n working machines, the LST of the sojourn time pdf TTR(t) is TTR (s) = N;1 X n=0 and the LST of TTR (n) (t) reads q N;n;1 TTR(n) (s) (3.8a) TTR(n) (s) = 2 ( + s)( + n) ; n( + s) TTR(n;1) (s) : (3.8b) With the aid of the k-th derivative of TTR (s) the k-th moment of TTR(t) can be determined. The variance Var[ TTR] is given by where Var[ TTR] = E[ TTR 2 ] ; E[ TTR] 2 = N;1 X n=0 00 TTR (n) (0) = n 3 00 TTR (n;1) (0) and k TTR (n) (0) = lim s!0 d k TTR (n) (s) s k : q N;n;1 00 TTR (n) (0) ; E[ TTR] 2 (3.9) +2 2 n 1 ; n 0 TTR (n;1) (0) o n ; n 0 TTR (n;1) (0) o It is clear that 0 TTR (n;1) (0) = ;E[ TTR (n;1) ] which is the mean time to repair given that n ; 1machines are working. Further it is 00 TTR (0) (0) = 2= Nonidentical failure and repair times For a system with only N = 2 machines we are able to generalize the results from above to the case of nonidentical failure and repair times. For systems with an arbitrary number of machines, the derivation of the variance of the time to repair seems to be quite tedious since one has to take into account not only the number of failed machines but also which ones are already broken down. For machine 1, the distribution function of the time to repair can be given analogously to the derivation above by TTR 1 (t) = q 1 1 B 1 (t) + q 1 0 B 1 (t) 1X k=0 P 1 [K = k] B (k) 2 (t) (3.10) 8

10 where q 1 i denotes the probability that machine 1 nds i machines at the repair facility upon its failure and B i (t) is the distribution function of the repair time of machine i. By means of the arrival theorem it can be shown that q 1 i are dened by the following equations: q 1 0 = and q 1 1 = 1 ; q 1 0 = : (3.11) The number of service preemptions is geometrically distributed with parameter p 1 which is given by p 1 = 2 =( ). As in the previous section the moments of the time to repair of machine 1 can be obtained from the LST of the pdf of sojourn times. It is: E[ TTR 1 ] = ( ) (3.12a) Var[ TTR 1 ] = : (3.12b) The moments of the sojourn time for machine 2 can simply be obtained by interchanging the indices 1 and 2ineqns. (3.12a) and (3.12b). Notice that in the case of a symmetric system eqn. (3.12a) resolves to eqn. (3.6) whereas eqn. (3.12b) resolves to eqn. (3.9). 4. Variance of sojourn times in the M=M=1==N{FCFS system By means of the LST it is very hard to derive the variance of sojourn times of the M=M=1==N{FCFS system directly from the waiting time pdf given in [12]. We propose an alternative computational method to gain this performance measure. Let TTR denote the sojourn time of an arbitrary machine. The number of machines already present at the repair station upon the arrival of an arbitrary machine is represented by the random variable L. In the case of identical service requirements for all machines, the arrival probability P [L = n] = q n can be simply obtained by the arrival theorem. Notice that in the FCFS case the product form property of the model gets lost with dierent service requirements. Given that there are L machines present at the repair station upon arrival, the TTR of an arbitrary machine consists of the forward recurrence time of the service time, B, of the machine currently being repaired, the repair time for L ; 1waiting machines, and its own repair time, i.e.: TTR = B + (L ; 1) B + B = B + L B = (L +1) B: (4.13) The nal step of this algebraic reshaping is obvious since repair times are exponentially distributed. Therefore B follows the same distribution as B. From this equation it 9

11 is clear that under stationary conditions the mean and the variance of TTR are as follows: TTR = (L +1) B (4.14a) Var[ TTR] = (L +1) Var[ B ] + Var[ L ] B 2 = (L +1+Var[ L ]) ;2 : (4.14b) The last step follows from the observation that B = ;1 and Var[ B ]= ;2. Eqns. (4.14a) and (4.14b) can be obtained by taking the generating function of the continuous time random variable (L + 1) B where L itself is a discrete-time random variable. A further alternative method to compute the variance of sojourn times is to approximate the (continuous) pdf of sojourn times by a discrete (see Tran-Gia [22]). This approximation method provides a control of the relative errors of the mean and the squared coecient of variation such that any accuracy (up to the processing accuracy) can be achieved. 5. Numerical Results In this section we will use the methods derived above to present numerical results for two congurations of the MIP. All results are shown versus the utilization of the repairman. In the rst system one operator (m = 1) serves eight machines (N = 8). All machines have identical breakdown and repair distributions ( i = =4, i = ) we will compare the queueing disciplines FCFS and LCFS-PR. The repair rate of the operator runs from =10to = 50 yielding trac loads ranging from 60 %to nearly 100 %. The mean of the sojourn time increases with growing utilization of the operator (Fig. 2). The plot is not linear and there is no dierence between the queueing disciplines FCFS and LCFS-PR because the order of repairing the machines has no inuence on the mean repair time (see Sec. 3). The variances of the sojourn time dier signicantly for the dierent queueing disciplines (Fig. 3). For LCFS-PR the variance of the sojourn time depends on how often the repair process is interrupted by a breakdown of another machine. There is no upper bound on the number of times one repair cycle can be interrupted. For FCFS the sojourn time of a machine is inuenced by another machine, only if this machine has broken down before the tagged machine. This leads to a higher variance of the sojourn time for LCFS-PR than for FCFS. For the system investigated here, the variances of the sojourn times for the two disciplines dier by a factor of approximately 100 and more. The dierence increases with higher utilizations. Higher utilizations mean a lower repair rate which leads to a higher probability that the service of a machine will be preempted (see eqn. (3.4)). In the second system we want to focus on the problem whether a machine with a high breakdown rate and short repair times behaves dierent from a machine with a small 10

12 breakdown rate and long repair times. One operator (m = 1) services two machines (N = 2). Breakdown and repair times are exponentially distributed with 1 = 70, 2 = 10, and 1 = 7 2. The queueing discipline at the service station is LCFS-PR. In this case the sojourn times in the repair station can not be compared because the repair times for the two machines are dierent. Therefore we will focus on the waiting times. Clearly, the mean time to wait is E[ TTW ] = E[ TTR] ; E[ B ] whereas the variance of the time to wait is Var[ TTW ] = Var[ TTR] ; Var[ B ]. E[TTR] Var[TTR] LCFS-PR FCFS ρ Fig. 2: Mean sojourn time ρ Fig. 3: Variance of the sojourn time Fig. 4 and 5 show a signicant dierence between the two machines. The repair of machine 1 with short breakdown and repair times is usually not interrupted by a breakdown of the other machine with higher breakdown and repair rates. The opposite is true for machine 2 because of its longer repair times. This leads to a higher mean and variance of the waiting time for machine 2. The mean and the variance of the sojourn time of machine 2 might be lower than those of machine 1 if the repair rates dier signicantly. E[TTW] machine 1 machine ρ Fig. 4: Mean waiting time Var[TTW] machine 1 machine ρ Fig. 5: Variance of the waiting time The analysis of this system with the queueing discipline FCFS is not possible with the help of the mean value analysis, because FCFS nodes with class dependent service rates have no product form. However, similar results can be expected. 11

13 6. Conclusion and Outlook In this paper we investigated the M=M=1==N{LCFS-PR system and presented a recursive denition of the sojourn time probability distribution function for it. By means of a transformation method we obtained the rst two moments of sojourn times in the case of an arbitrary number of identical machines and identical repair requirements. This result was generalized regarding two heterogeneous messages with dierent service requirements. Further, we derived a simple expression for the variance of sojourn times of the M=M=1==N{FCFS system. Finally, we compared numerical results of the variance of sojourn times for the queueing disciplines FCFS and LCFS-PR. The results show that the LCFS-PR repair strategy leads to a considerable increase of the variance of sojourn times as compared to the FCFS case. However, if one is interested in the real eect of service preemption on sojourn times in the LCFS-PR case, one has to nd either an explicit expression for the rst two moments from the Laplace-Stieltjes-Transform of the sojourn time probability distribution function of the M=M=1==N{LCFS system given in [21], or one has to implement a numerical procedure to compute these moments numerically. Since the steady stationary probabilities are the same for FCFS and LCFS-PR, it might be interesting to investigate the magnitude of dierence among the variances of sojourn times for these queueing disciplines in the case of both generally distributed failure and repair times. Acknowledgement The authors would like to thank Prof. P. Tran-Gia for the support of this work and for his eorts to review the paper. The stimulating discussions with N. Gerlich are greatly appreciated. References [1] Arnold O. Allen. Probability, Statistics, and Queueing Theory. Academic Press, London, 2nd edition, [2] Forest Baskett, K. Mani Chandy, Richard R. Muntz, and Fernando G. Palacios. Open, closed and mixed networks of queues with dierent classes of customers. Journal of the ACM, 22(2):248{260, April [3] F. Benson and D. R. Cox. The productivity of machines requiring attention at random intervals. Journal of the Royal Statistical Society, B, 13:65{82, [4] John W. Boyse and David R. Warn. A straightforward model for computer performance prediction. ACM Comput. Surv., 7(2):73{79, June [5] John A. Buzacott and J. George Shanthikumar. Stochastic models of manufacturing systems. Prentice-Hall, Englewood Clis, NJ, [6] K. Mani Chandy and A. J. Martin. A characterization of product-form queueing networks. Journal of the ACM, 30(2):286{299, April

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