INTRODUCTION

This study considers a system with K-station in series, supposes that arrivals at station 1 are generated from an infinite population according to a Poisson distribution with mean serviced units will move successively from one station to the next until they are discharged from station K. Service time at each station I has exponential distribution. The problem of point estimation is formulated as follows: A random sample is drawn from a population whose distribution has a known mathematical form but involves a certain number of unknown parameters. On the basis of this random sample, it is required to estimate the value of unknown parameters of the population.

Research has done on parameter estimation of the extended generalized Gaussian family distributions using maximum likelihood scheme (Lee, 2010). Time domain DS-UWB channel maximum likelihood algorithm was in working by Yin et al. (2011) but the work of Shi et al. (2011) was in solving project scheduling problems using estimation of distribution algorithm with local simplex search. Desta et al. (2011) carried out review on selected channel estimation algorithms for orthogonal frequency division multiplexing system. Wang et al. (2011) deal with a novel joint estimation algorithm for multi-parameter of underwater acoustic channels. Abd El-Salam (2011) deals with an efficient estimation procedure for determining ridge regression parameter. Ojo (2010) has work done in on the estimation and performance of one-dimensional autoregressive integrated moving average bilinear time series models. Oyeyemi et al. (2010) consider on the estimation of power and sample size in test of independence. Ojo (2008) carried out on the performance and estimation of spectral and bispectral analysis of time series data.

DESCRIPTION OF THE PROBLEM AND ITS SOLVING

Under these conditions, the out put of station I follows Poisson process with arrival rate λ, where station I follows the model (M/M_i/1):(GD/∞/∞).

This means for the ith station, the steady state probabilities are given by:

For I = 1, 2,…, k, where n_i is the number in the system consisting of station I only. Steady state results will exist only if utilization factor ρ_i = λ/μ_i<1.

To find the likelihood function, it is necessary to find three basic components, Namely:

•	The initial number of units ni with probability Pr (ni) = for I = 0, 1, 2,…, k
•	The inter-arrival time of lengths t1 for R arrival units with probability
•	The service time of durations qj for M units with probability:

It follows that the likelihood function is given by:

Hence:

Hence:

Where:

After differentiating we get:

(1)

And:

(2)

where, and are the maximum likelihood estimators of the parameters λ and μ_i, respectively. Then:

Where:

Using the Eq. 1 and 2, then:

(3)

And:

(4)

where, is the maximum likelihood estimator of E (n_i).

From the Eq. 3 and 4 we get:

(5)

And:

(6)

Substituting for its value in the Eq. 6, we get the equation:

Where:

Hence:

(7)

Suppose that M = R = n_i and using the Eq. 5 and 7, it follows that:

And:

In addition, we can obtain:

Table 1:	Application with numerical results*
*Suppose that the following data access times of units were given

In Table 1, we can obtain the following results: