s. The method arrive during a customer service is given by Equation 3.28. can be applied to both equilibrium and transient queues (Takacs, 1962). (3.28) The most commonly used transforms are the probability genera- ting function (z-transform), for discrete random variables and the Laplace transform for non-negative continuous random variables. If Transforming to get the generating function for the PR's gives N is a discrete random variable and Ta non-negative continuous (or Equation 3.29. mixed) random variable, Equation 3.25 may be obtained, where Pk is given by Equation 3.26 and H(t) is the distribution of T. =Ele"I=J" (3.25) (3.29) Pk=Pr|N=k (3.26) Figure 3.9 The embedded Markov chain of the M/G/1 queue 3/10 Queuing theory The interchange of summation and integral can readily be justi- Q(z), as derived, is the generating function for the number of fied. Thus the transform of p is obtained through the Laplace customers left behind in queue by a departing customer. However transform of H. The embedded Markov chain of arrivals is deter- this is then the equilibrium distribution for the number of customers mined through p but as the reader can see from Figure 3.9 the empty which the customer finds on arrival. This follows from the equality state is a special case. A customer departing and leaving the queue of equilibrium customer arrival distributions and customer depar- empty marks the start of an idle period. ture distributions result described earlier. Note also that Q gives the The simplest way to analyse this chain is to work with the random time equilibrium of customers in queue as well because of Poisson variables directly. (Alternatively transform the equilibrium equa- arrivals. tions of the embedded Markov chain). Let Q denote queue length Q can be used to obtain the equilibrium waiting time transform. immediately following the nth departure, A the number of arrivals Let S be the total time a customer spends in the system (queuing during the nth service then Equation 3.30 can be obtained, where I time plu