Statistical Analysis for a Three Service Point Tandem Queue with Blocking

A maximum likelihood estimator (MLE), a consistent asymptotically normal (CAN) estimator and asymptotic confidence limits for the expected number of entities in the system in a three service point tandem queue with blocking and busy service point 1 and zero queue capacity in front of service points 2 and 3 are obtained.


Introduction
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 statistical inference problems.Nowadays, statistical analysis of queueing systems is an important area of research in Queueing theory.Parametric estimation, Interval estimation and Bayesian estimation are some of the essential tools to understand any random phenomena using stochastic models.Analysis of queueing systems in this direction has not received much attention in the past.Whenever the systems are fully observable in terms of their random components such as interarrival times and service times, standard parametric techniques of statistical theory are quite appropriate.Recently, Narayan Bhat (2003) has provided an overview of methods available for estimation, when the information is restricted to the number of entities in the system at some discrete points in time.Narayan Bhat (2003) 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 queues.Table 1 indicates the present state of work of queueing systems, wherein the asymptotic confidence limits for measures of system performance are obtained.
Generally speaking, the queueing models assume that each service channel consists of only one service point.Situations do exist, where each service channel may consist of several service points in series.In this situation, an entity must pass through all these service points in succession before completing its service.Such situations are known as queues in series or tandem queues.e.g., (a) in a manufacturing process, units must pass through a series of service points (work stations), where each service point performs a given task or job, (b) in a university registration process, each registrant must pass through a series of counters such as advisor, department chairman and cashier, (c) in a clinical physical examination procedure, a patient must go through a series of stages such as laboratory tests, ECG and chest X-ray.In all these models, it is not only sufficient to know how many entities are there in the system but also where they are.In general, in all the above mentioned tandem queueing models studied so far, it is assumed that no queues are allowed in front of service points.This is not so in any real life situation.Hence, an attempt is made in this paper to study in detail a three service point tandem queue, where service point 1 is busy but no queue is allowed at service point 2 and service point 3.A MLE, CAN and asymptotic confidence limits for the expected number of entities in the system are also obtained.

System Description and Assumptions
Consider a simplified one channel queueing system consisting of three series service points as below : An entity arriving for service must pass through service point 1, service point 2 and service point 3 before completing its service.The assumptions of the model are as follows: i. Service times at service point 1, service point 2 and service point 3 are independent and exponentially distributed with service rates ii.Service point 1 is busy but no queue is allowed at service point 2 and service point 3.
iii.Service point 2 and service point 3 are either free or busy.
iv. Service point i (i=1, 2) is said to be blocked, if the entity in service point i(i=1, 2) completes its service before service point (i+1) (i=1, 2) becomes free.In this case, the entity cannot wait between the service points i and (i+1) (i=1, 2), since this is not allowed and the entity remains in service point i( i=1, 2) itself.

Analysis of the System
Let the symbols 0, 1, and b represent free, busy and blocked states of a service point.Let X(t), Y(t) and Z(t) respectively denote the states of service point 1, service point 2 and service point 3 and the vector process the state of the system at time t.Since the interarrival and service times are all exponential, it clearly follows that the process W(t) is a Markov process with the infinitesimal generator given by represent the probability that the system is in state   k j, i, at time t with the initial condition   1 0 p 100  .From the infinitesimal generator given in (3.2), we have the following system of differential -difference equations:

Steady state solution
The equations (3.3) -(3.10) can be solved using the fact that Since we wish to study the stationary behaviour of the system, let

Expected number of entities in the system
The expected number of entities in the system is given by Further,   In the next section, we obtain the MLE and CAN for the expected number of entities in the system.

Let
. Clearly, the expected number of entities in the system given in (3.21) yields By applying the multivariate central limit theorem given above, it is readily seen that where σ ˆis a consistent estimator of   θ σ .

Numerical Illustration
To study the performance of S L ˆ, random samples of 5000 observations are generated independently 50 times from exponential distributions assuming θ 1 = 2, θ 2 = 3 and θ 3 = 4.Using the generated samples, the Maximum Likelihood estimates of θ 1 , θ 2 and θ 3 are obtained.The estimated value of S L i.e., S L ˆ is then obtained using these estimates.Table 2 shows the calculated values of S L and S L ˆand the corresponding bias which is defined as the difference between the value of S L ˆand S L .
The performance of s L ˆis measured in terms of Mean Square Error that is defined as Mean Square Error =   .The Mean Square Error value is obtained as 0.002014933.It is to be noted that values of Mean Square Error of any estimator close to zero indicate that the estimator is good.Thus, based on the numerical results, it is reasonable to conclude that the proposed estimator performs well.

3.2 A CAN estimator using the multivariate central limit theorem
3.