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1 International Mathematical Forum, Vol. 6, 2011, no. 62, Estimating Changes in Traffic Intensity for M/M/1/m Queueing Systems Sarat Kumar Acharya Departament of Statistics, Sambalpur University Jyoto Vihar , Sambalpur, Orissa, India César Emilio Villarreal-Rodríguez Facultad de Ingeniería Mecánica y Eléctrica Universidad Autónoma de Nuevo León A. P. 66-F, Ciudad Universitaria San Nicolás de los Garza, N.L., Mexico cesar.villarrealrd@uanl.edu.mx Abstract In this note we give a formula for estimating the number of arrivals for which the traffic intensity changes in an M/M/1/m queueing sistems. Mathematics Subject Classification: 60K25, 68M20, 62M05. Keywords: change point, maximum likelihood estimation, M/M/1/m queue, traffic intensity. 1 Introduction Change is a natural phenomenon which occurs in every sphere of works. An awareness of these changes can help people to avoid unnecessary losses and to harness beneficial transitions. In many practical situations a statistician is faced with the problem of detecting the number of change points and their locations. A change point is a place or time point such that the observations follow one distribution up to that point and follow another distribution after that point. So the change point problem is twofold: one is to decide if there is any change hypothesis testing problem, another is to locate the change point when there is a change present estimation problem. Change point models have been one of the main research topics in statistics for many

2 3088 S. K. Acharya, C. E. Villarreal-Rodríguez years. In view of it applicability, it has received great notoriety over the last six decades in the statistical literature. The problem of testing of parameter change has long been a core issue in statistical inferences. It originally started in the quality control context and then rapidly moved to various areas such as economics, finance, transportation systems, statistical quality control, inventory, production processes, communication networks and queueing, control problems, medicine. Since the paper of Page [9], the problem has generated much interest and a vast amount of literatures have been published in various fields. For a general review, we refer to Csorgo and Horvath [4], Chen and Gupta [2] and the articles therein. The change point problem was first dealt in i.i.d samples but it became very popular in time series models since the structural change often occurs in economic models owing to a change of policy and critical social events. For relevant references in i.i.d. samples and time series models, we refer to Hinkley [5], Brown, Durbin and Evans [1], Zacks [12], Picard [10], Csorgo and Horvath [3], Krishnaiah and Miao [7], Lee and Park [8], and the references therein. To detect the number of change points and their locations in a multidimensional random process Vostrikova [11] proposed a method known as the binary segmentation procedure. It has the merits of detecting the number of change points and their positions simultaneously and saving a lot of computation time. In 1995, Jain [6] gave a formula for estimating the number of customers from which change the traffic intensity according to the number of present customers in a M/M/1/m queueing system, despite he said that the interarrival time distribution is assumed to change at some time before the mth arrival. In this note we assume that the traffic intensity changes after an unknown number of arrivals and we shall use similar technics to that given in [6] in order to estimate the traffic intensities after and before the change, and the number of arrivals in which the traffic intensity change, according to the maximum likelihood estimation. It is proposed a single service G/M/1/m queueing system, where m is the capacity of the system, that is, the space state of the system is E = {0, 1, 2,...,m}; X n is the number of customers present in the system just before the time T n of the nth arrival; μ is the rate of service and A is the interarrival time distribution. So X =X n n=1 is a Markov chain with state space E. The probability transition matrix P is given by

3 Estimating Changes for M/M/1/m 3089 P = 1 b 0 b b k b 1 b b k b 2 b m 1 b k b m 1 b m 2 b 1 b 0 1 m 1 b k b m 1 b m 2 b 1 b 0, 1 where b n is the probability of occurring n service completions during an interval time when there is at least n customers in the system, and it is given by b n = 1 n! + μt n expμtdat, for n {0, 1, 2,...}. 2 0 In the case of an M/M/1/m system, At =1 exp λt for some λ>0 and b k = for k {0, 1,...,m 1}, 1 + k+1 where is the traffic intensity which is give by = λ μ. Suppose that we observe the Markov chain X with transition probability matrix P until the total number of arrivals reaches a preassigned fixed value M. Let L M be the number given by L M = PrX 0 = i 0 M PrXk = i k Xk 1 = i k 1, 3 k=1 where i k E for k {0, 1,...M}. If we omit the first factor in 3 and we take logarithm, then taking l =lnl M, we have l = m + m 1 n m 1,j + n m,j lnb m j +n m,0 + n m 1,0 ln m 2 i=j 1 n i,j ln b i j+1 + m 2 n i,0 ln 1 i b k, where n i,j denotes the number of transitions from i to j. 1 m 1 b k 4

4 3090 S. K. Acharya, C. E. Villarreal-Rodríguez 2 Estimation of Change-point Assume that we have an M/M/1/m queueing system in which the traffic intensity changes just after τ arrivals. So the ith interarrival time distribution A i is given by 1 exp λ 1 t for i {0, 1,...,τ} A i t =. 5 1 exp λt for i {τ,τ +1,...,M} We have two transition matrices Pb and P a for the system: The first one works through the first τ arrivals, and the second one works just after the τth arrival to the Mth arrival. The matrices Pb and Pa are like P given in 1 but changing b k by b b k and ba k respectively. For each transition matrix, we have and b b k = 1, for k {0, 1,...,m 1}, k+1 b a k =, for k {0, 1,...,m 1}, k+1 where 1 = λ 1 μ 1 and = λ are the traffic intensities of the system before and μ after the change point respectively. So next formula express the log likelihood function and was used to estimate, 1 and τ. l, 1,τ = m n b m 1,j + nb m,j ln m j+1 +n b m,0 + nb m 1,0 ln 1 m 1 + m 1 m 2 i=j 1 + m 2 n b i,0 ln n b i,j ln 1 i i j k k+1 1+ m j+1 + m n a m 1,j + n a m,jln +n a m,0 + na m 1,0 ln 1 m 1 + m 1 m 2 i=j 1 + m 2 n a i,j ln n a i,0 ln 1 i 1+ i j+2, 1+ k+1 1+ k+1 where n b i,j and n a i,j denote the number of transitions from i to j before and after the change point respectively. Note that n b i,j and n a i,j depend of the value of τ. 7

5 Estimating Changes for M/M/1/m 3091 In order to get the maximum likelihood estimator of conditional on τ = c we derive partially equation 7 to get m n a m,j + m 1 i+1 l, 1,c = m n a m,j n a i,j m j+m 1 i+1 n a i,j i j +1. From 8 we have that the maximum likelihood estimator for is 8 ˆ = m n a m,j m n a m,j + m 1 i+1 n a i,j 9 i+1 m j+m 1 n a i,j i j +1 and analogously, themaximum likelihood estimator for 1 is ˆ 1 = m n b m,j m n b m,j + m 1 i+1 n b i,j. 10 i+1 m j+m 1 n b i,j i j +1 To obtain the maximum likelihood estimator of τ, we can evaluate exhaustively equation 7, from τ =1until τ = M 1, with the corresponding values of ˆ and ˆ 1 given in 9 and 10. ACKNOWLEDGEMENTS. The research was conducted while the first author was visiting the FIME, UANL, Mexico. Thanks are due to CONACyT for supporting the research under grant number and PAICyT under grant number CE References [1] R.L. Brown, J. Durbin and J.M. Evans, Techniques for testing the constancy of regression relationship over time, Journal of Royal Stat. Soc., series B, , [2] J. Chen and A.K. Gupta, Parametric Statistical Change Point Analysis, Birkhäuser, Boston, [3] M. Csorgo and L. Horvath, Nonparametric methods for change point problems. In: P.R. Krishnaiah, C.D. Rao Eds, Hanbook of Statistics, Vol. 7, pp , Elsevier, New York, 1988.

6 3092 S. K. Acharya, C. E. Villarreal-Rodríguez [4] M. Csorgo and L. Horvath, Limit Theorems in Change Point Analisys, Wiley, NewYork, [5] D.V. Hinkley, Inference about the change point from cumulative sums test, Biometrica, , [6] S. Jain, Estimating changes in traffic intensity for M/M/1 queueing systems. Microelectron. Reliab., , [7] P.R. Krishnaiah, and B.Q. Miao, Review about estimation of change points. In: P.R. Krishnaiah, C.D. Rao Eds, Hanbook of Statistics. Vol. 7 pp , Elsevier, New York, [8] S. Lee, and S. Park. The cusum of squares test for scale change in infinite order moving average processes, Scandinavian Journal of Statistics, 28, 2001, [9] E.S. Page, A test for change in a parameter occurring at an unknown point, Biometrica, 42, 1955, [10] D. Picard, Testing and estimating change-points in time series, Advances in Applied Probability, 17, 1985, [11] L.Y Vostrikova, Detecting disorder in multidimensional random processes, Soviet Math. Dokl., , [12] S. Zacks, Survey of classical and Bayesian approaches to the change point problem: fixed sample and sequential procedures of testing and estimation. In: M.H. Rivzi et al. Eds. Recent Advances in Statistics, pp , Academic Press, New York, Received: June, 2011