Example of Single servers

(M/M/1):(GD//)

(M/M/1):(FCFS//)

Let us denote the following

•    denotes the nth customer in the system at time t
  

• number of arrivals in (0, t) time interval
  

• number of departures in (0, t) time interval
  

•   number of customer in the system at time t, i.e.,
  

• the remaining time required to empty the system of all customers present at time t
  

• , i.e., average arrival rate in the interval (0, t)
  

•  accumulated customer seconds upto time t
  

• , i.e., system time per customer averaged over all customer in the queueing system in the interval (0, t)
  

•  for
  

• inter arrival time for , and as per our basic assumption we have
  
  
  

•   for and as per our assumption we have
  

• for
  

• total time spent by , this is also known as the system time, i.e., the time spent in the system or queue by