Confidence limits for expected waiting time of two queuing models

A maximum likelihood estimator (MLE), a consistent asymptotically normal (CAN) estimator and asymptotic confidence limits for the expected waiting time per customer in the queues of M |M |1|∞ and M |M |1|N are obtained.


Introduction
Parametric estimation is one of the essential tools to understand the random phenomena when using stochastic models.Whenever the systems are fully observable in terms of their basic random components such as inter-arrival times and service times, standard parametric estimation techniques of statistical theory are quite appropriate.Most of the studies on several queueing models are confined to only obtaining expressions for transient or stationary (steady state) solutions and do not consider the associated inference problems.Recently, Bhat (2003) has provided an overview of methods available for estimation, when the information is restricted to the number of customers in the system at some discrete points in time.Narayan Bhat has also described how maximum likelihood estimation is applied directly to the underlying Markov chain in the queue length process in M |G|1 and GI|M |1.The MLE, CAN and asymptotic confidence limits for the expected waiting time per customer in the queues of M |M |1|∞ and M |M |1|N , are obtained in this paper.In the following section, these two models and the expected waiting time per customer for each model are explained briefly.

Model I (M |M |1) : (F CF S|∞|∞) queue
It can be readily seen [1] that the difference-differential equations governing M |M |1 are given by As t → ∞, the steady state solution can be proved to exist, when λ < µ.Assuming that p ′ n (t) → 0 and p n (t) → p n as t → ∞, for n = 0, 1, 2, . . ., we have Solving these difference-differential equations, we have where ρ = λ µ < 1. Clearly (1) corresponds to the probability mass function of the geometric distribution and it can easily be shown that the expected waiting time per customer in the queue is given by Model II (M |M |1) : (GD|N |∞) queue This model is essentially the same as Model I, except that the maximum number of customers in the system is limited to N (maximum queue length is N − 1) [1].The steady state equations for this model are given by The solution of these difference-differential equations is given by The expected number in the system is given by Since there is a limit on the queue length and some customers are lost, it is necessary to compute the effective arrival rate λ eff , which is given by λ eff = λ(1 − p N ).Further, it can be shown that the expected number of customers in the queue is Hence, the expected waiting time per customer in the queue is given by 3 MLE and CAN estimator for the expected waiting time

Model I
The average waiting time per customer in the queue, given in (2), reduces to and hence the MLE of W Q is given by

Model II
The average waiting time per customer in the queue, given in (3), reduces to . (5) and hence the MLE of W Q is given by It may be noted that i ŴQ , given in ( 4) and (6), are real valued functions in Xi and Ȳi , i = 1, 2, which are also differentiable.Consider the following application of multivariate central limit theorem [3].

CAN Estimator
Model I By applying the multivariate central limit theorem [4], it readily follows that Hence, 1 ŴQ is a CAN estimator of 1 W Q .There are several methods for generating CAN estimators -the Method of Moments and the Method of Maximum likelihood are commonly used to generate such estimators [4].

Model II
As in Model I, here too, we have , where θ = (θ 1 , θ 2 ), and where 2 W Q and 2 ŴQ are given by ( 5) and ( 6) respectively.Further, 4 Confidence limits for expected waiting time Let i σ 2 ( θ) be the estimator of i σ 2 (θ) (with i = 1, 2 representing Models I and II) obtained by replacing θ a consistent estimator where k α 2 is obtained from normal tables.Hence, a 100(1 − α)% asymptotic confidence interval for i W Q is given by

Numerical Results
As is to be expected, W q is an increasing function of λ, and a decreasing function of µ, for both M |M |1|∞ and M |M |1|N queuing systems [See Tables 1 and 2].
Let X i1 , X i2 , ..., X in (with i = 1, 2 representing Models I and II) be random samples of size n, each randomly drawn from different exponential inter-arrival time populations with the parameter λ.Also, let Y i1 , Y i2 , ..., Y in (with i = 1, 2 representing Models I and II) be random samples of size n, each drawn from different exponential service time populations with the parameter µ.It is clear that E( Xi ) =1λ and E( Ȳi ) = 1 µ , where Xi and Ȳi , i = 1, 2, are the sample means of inter-arrival times and service times respectively corresponding to Models I and II.It can be shown that Xi and Ȳi (with i = 1, 2 representing Models I and II) are the MLEs of 1 λ and 1 µ respectively.Let θ 1 = 1 λ and θ 2 = 1 µ respectively.

Table 2 :
M |M |1|N : F CF S with 99% confidence interval and sample size of 20.