Introduction to Waiting Line ManagementMAH2021In the birth-death processes, there is only one state variable,Q, namely, the number ofelements in the population or in the line. Both birth and death rates depend onQ. IfQ=i,the birth rate isiand the death rate is.iSince, a birth increases the population by 1,the rate at whichQincreases fromitoi+1 isi( the probability of increasing thepopulation). Similarly,i(the probability of decreasing the population) is the rate atwhich the populationQdecreases fromitoi–1.The states 0, 1, 2, 3, … represent theincrease in population by 0, 1, 2, 3, … respectively.So, the only non-zero transition ratesarei(when the population increases fromitoi+ 1) andi( when the populationdecreases fromitoi–1). Transition matrix of the birth-death processes is shown below...3210...3210...........................000...000...000...00003322110Rates always presume the existence of events, such as arrivals or at least changes of state.Consequently, there can be no rate to go from stateito statei. There is, however, a rateof leaving statei. This rate is equal to the sum of the rates of going from statesitoi-1 ori+1, that is,.iiIn the literature, leaving rate of stateiis defined by–().iiTheresulting transition matrix is then...3210A =...3210...................................000...0)(.00)(...000332222111100Note that the sum of the probabilities of changing from a state and changing to that stateis equal to zero. If),,...,,,(1210iibe the probability that the system is in statei, then for steady-state probability (probability of any state that remains the same), thesteady state equations will be0A