Maximum likelihood estimates and confidence intervals of an M/M/R/N queue with balking and heterogeneous servers

This paper considers an M/M/R/N queue with heterogeneous servers in which customers balk (do not enter) with a constant probability (1-b). We develop the maximum likelihood estimates of the parameters for the M/M/R/N queue with balking and heterogeneous servers. This is a generalization of the M/M/2 queue with heterogeneous servers (without balking), and the M/M/2/N queue with balking and heterogeneous servers in the literature. We also develop the confidence interval formula for the parameter rho, the probability of empty system P-0, and the expected number of customers in the system E[N], of an M/M/R/N queue with balking and heterogeneous servers. The effects of varying b, N, and R on the confidence intervals of Po and E[N] are also investigated

Maximum Likelihood Estimates and Confidence Intervals of an M/M/R/N Queue with Balking and Heterogeneous Servers

這篇論文探討M/M/R/N排隊系統，包括相異的服務者，以及顧客遇到阻礙（不進入排隊）的機率為常數 。此篇論文含蓋(i)相異服務者、沒有阻礙條件的M/M/2排隊系統；和(ii)相異服務者、有阻礙條件的M/M/2/N排隊系統。 我們首先推導此排隊系統參數 ， ， 的最大概似估計。然後，我們也分別推導系統參數 、系統無顧客時的機率 、以及系統內顧客數的期望值之信賴區間公式。而變數 ， ，及 在 信賴區間上的影響也有深入探討。CONTENTS Abstract 1 Introduction………………………………………………………3 1.1 Problem statements……………………………………………4 1.2 Literature review……………………………………………4 2 M/M/R/N Queue with Balking and Heterogeneous Servers..6 2.1 Likelihood function…………………………………………8 2.2 Maximum likelihood estimates……………………………11 2.3 Special cases…………………………………………………11 3 Confidence Interval Formulas for , and ……………12 3.1 Confidence interval for …………………………………12 3.2 Confidence interval for …………………………………17 3.3 Example…………………………………………………………21 4 Conclusions and Future Research……………………………22 4.1 Conclusions……………………………………………………22 4.2 Future research ……………………………………………22 References……………………………………………………………23 Appendix I……………………………………………………………24 Appendix II……………………………………………………………2

Maximum likelihood estimates and confidence intervals of an M/M/R/N queue with balking and heterogeneous servers

This paper considers an M/M/R/N queue with heterogeneous servers in which customers balk (do not enter) with a constant probability (1 - b). We develop the maximum likelihood estimates of the parameters for the M/M/R/N queue with balking and heterogeneous servers. This is a generalization of the M/M/2 queue with heterogeneous servers (without balking), and the M/M/2/N queue with balking and heterogeneous servers in the literature. We also develop the confidence interval formula for the parameter ρ, the probability of empty system P0, and the expected number of customers in the system E[N], of an M/M/R/N queue with balking and heterogeneous servers. The effects of varying b, N, and R on the confidence intervals of P0 and E[N] are also investigated