Estimating Change Point in Single Server Queues

The paper is concerned with the study of change point problem in the inter-arrival time and service time of single server queues. Maximum likelihood estimators of the parameters are derived. A test statistics has been developed and its properties have been studied.


Introduction
Statistical inference plays a major role in any use of a queueing models in decision making. Many authors have studied the parameter estimation problem in queueing models. Basawa and Prabhu [4,5] have discussed moment and likelihood estimation of the model parameters for single server queues using various sampling plans. Bhat and Rao [6] have provided an exhaustive survey of results on inference for queueing systems. Basawa and Bhat [2] studied sequential inference for the parameters of a GI/G/1 queue. An empirical Bayes approach was used for estimating by Thiruvaiyaru and Basawa [15]. Basawa et al. [3] presented a maximum likelihood method for estimating the parameters of the arrival and service time distribution using only the information on the waiting times of customers in GI/G/1 queue with "first come first served" queue disciline. Clarke [10] obtained the maximum likelihood estimates for the arrival and service parameters of an M/M/1 queue. A review of the literature on the subject reveals that so far only single server queues have been considered from an inferential viewpoint. crane and lemoine [14] have applied simulation techniques to the problem of estimating the steady state mean waiting time in a single server queue. Acharya [1] have discussed the rate of convergence of the distribution of the maximum likelihood estimators of the arrival and the service rates in a GI/G/1 model.
Change is a natural phenomenon which occurs in every sphere of works. In statistics we are interested in the statistical analysis of change point detection and estimation. Let X 1 , X 2 , X 3 , . . . , X n be a sequence of independent random variables with probability distribution functions F 1 , F 2 , F 3 , . . . , F n , respectively.
Then, in general, the change point problem is to test the following null hypothesis: . . = F n verses the alternative hypothesis . . , F n belongs to the common parametric family F (θ), where θ ∈ R p , then the change point problem is to test the hypothesis about the population parameter θ i , i = 1, 2, 3, . . . , n, H 0 : θ 1 = θ 2 = . . . = θ n = θ(unknown) verses the alternative hypothesis 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. The change point problem was first dealt in independent and identically distributed samples but it became very popular in time 20 Estimating Change Point in Single Server Queues series models. For relevant references in i.i.d samples and time series models, we refer to Brown, Durbin and Evans [7], Wichern, Miller and Hsu [17], Zacks [18], Krishnaiah and Miao [13] and the references therein.
The problem of estimating change point of the inter-arrival time distribution in the queueing is of great interest. Besides maximum likelihood and least square estimates, the Bayesian method is also a very useful technique for estimating parameters. Chernoff and Zacks [8] has studied the change point problem using the Bayesian method.
The main goal of this paper is to study the change point problem for the single server queue. In section 2 preliminary results about the maximum likelihood estimators of a GI/G/1 queue have been mentioned. Section 3 deals with the change point estimation for the interarrival time distribution of the GI/G/1 queue . A test statistic has been developed.
2 The GI/G/1 Queue Consider a single server queueing system in which the interarrival times {u k , k ≥ 1} and the service times {v k , k ≥ 1} are two independent sequences of independent and identically distributed nonnegative random variables with densities f (u; θ) and g(v; ϕ), respectively, where θand ϕ are unknown parameters. Let us assume that f and g belong to the continuous exponential families given by It is further assumed that the densities in (1) and in (2) are equal to Zero on (−∞, 0).
For simplicity we assume that the initial customer arrives at time t = 0. Our sampling scheme is to observe the system over a continuous time interval [0, T ] where T is a suitable stopping time. The sample data consist of where A(T ) is the number of arrivals and D(T ) is the number of departures during (0, T ]. Obviously no arrivals occur during some possible stopping rules to determine T are given below: Rule 1. Observe the system until a fixed time t. Here T = t with probability one and A(T ) and D(T ) are both random variables.
Rule 2. Observe the system until d departures have occured so that D(T ) = d.
Rule 3. Observe the system until m arrivals take place so that A(T ) = m. Here T = u 1 + u 2 + u 3 + · · · + u m and D(T ) are random variables. Under rule 4, we stop either with an arrival or in a departure. If we stop with an arrival, then The likelihood function based on data (3) is given by where F and G are distribution functions corresponding to the densities f and g, respectively. The likelihood function L T (θ, ϕ) remains valid under all the stopping rules.
The maximum likelihood estimates obtained from (5) are asymptotically equivalent to those obtained from (4) provided the following two conditions are satisfied as T → ∞:

Approximate maximum Likelihood Estimates
The interarrival time density f (u; θ) and the service time density g(v; ϕ) belongs to exponential families given by (1) and (2). It is easily verified that the moment generating function of the random variables h 1 (u) and h 2 (v) is given by The approximate likelihood function L a T (θ, ϕ) is reduced to and the log likelihood function becomes Differentiating l(θ, ϕ) a w.r.t θ and ϕ and then equating to zero, we get and From now on we shall write l for L a T . The estimating equation reduce to The solution for θ and ϕ from (14) and (15) are given bŷ

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Estimating Change Point in Single Server Queues

Change Point Problem of the Inter-arrival Time
Let's consider the GI/G/1 queueing system in which interarrival times u k , k ≥ 1 and the service times v k , k ≥ 1 are the two independent sequence of independent and identically distributed non-negative random variables with density f (u, θ) and g(v, ϕ) respectively.
We can write this as Following Kander and Zacks [11] we derive a test statistic using quasi Bayesian approach by considering the point of change l as a realization of a random variable L with a uniform prior density.
The loglikelihood function under the null hypothesis is given by Under the alternative hypothesis Now by taylor expansion as δ → 0 The likelihood ratio is as δ → 0 Now our test statistic is The exact distribution of T A(T ) is obtained as follows (Since h 1 (u i )'s are exponential under null hypothesis)

V ar[T A(T ) ] = var
Estimating Change Point in Single Server Queues The test statistics T A(T ) follows gamma distribution with mean 0 and variance E{A( Similarly we can show that when there is a change in parameter ϕ, then the test statistics is The test statistics T D(T ) follows gamma distribution with mean 0 and variance E{D(T )(D(T )−1)(2D(T )−1)} 6 · σ 2 2 (ϕ 0 )

Example
Let's consider the M/M/1 queueing system with a poisson arrival and exponential service time .Let the interarrival times u i , i ≥ 1 and the service times v i , i ≥ 1 are two independent sequence of independent and identically distributed non-negative random variables with density f (u, λ) and g(v, µ) respectively given as We are interested in testing the null hypothesis that u 1 , u 2 , · · · · · · , u A(T ) are i.i.d from exponential distribution with parameter λ 0 against the alternative hypothesis that at some point l a change occurs in parameter λ i.e. for some l ∈ 1, 2, 3, · · · · · · , A(T ) − 1, u 1 , u 2 , · · · · · · , u l are i.i.d from exponential distribution with parameter λ 0 and u l+1 , u l+2 , · · · · · · , u A(T ) are i.i.d from exponential distribution with parameter (λ 0 + δ).
We can write this as Then the likelihood function under the null hypothesis can be obtained as and the maximum likelihood estimator of λ isλ Under the alternative hypothesis the likelihood function The likelihood ratio is Now our test statistic is The exact distribution of T A(T ) is obtained as follows Similarly we can show that when there is a change in parameter µ, then the test statistics is