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In this paper, consider a single server queue in a hospital environment whose service time is governed by a Markov process. It is possible that the server changes its service speed many times while serving a patient. Here we have studied the order statistics for waiting time distribution where the probability density function of single order statistics $\phi_{i:n}$, cumulative density function of $\Phi_{i:n}$, joint probability density function of $\phi_{i:n}$ and $\phi_{j:n}$, probability density function of extreme order statistics. Also have been considered the moments and recurrence relation of order statistics, the probability density function of sample range and sample median. We derive minimum and maximum order statistics of the service time of patients in the system using first step analysis to obtain an insight on the service process. Further, we use order statistics to compute performance measures such as average queue length and waiting time for severe diseases especially in the outpatient department. This result effectively establishes that as the number of server increases, then the utmost and the minimum waiting time of the patients decreases. Also illustrate the application of the simple Markovian model by using real hospital data.

Estimation of the waiting time of patients in a hospital with simple Markovian model using Order Statistics