6.3.2 Patients arrive at a hospital emergency room according to a Poisson process of rate . The patients are treated by a single doctor on a first come, first served basis. The doctor treats patients more quickly when the number of patients waiting is higher. An industrial engineering time study suggests that the mean patient treatment time when there are k patients in the system is of the form mk D ????k=.kC1/, where and are constants with > > 0. Let N.t/ be the number of patients in the system at time t (waiting and being treated). Argue that N.t/ might be modeled as a birth and death process with parameters k D 

for k D 0;1; : : : and k D k=mk for k D 0;1; : : : : State explicitly any necessary assumptions.