Stochastic Processes with Balking in the Theory of Telephone Traffic

Many results in telephone traffic theory (and elsewhere) may be unified by Ihc introduction of balking. A call is said to balk if for any reason it refuses service on arrival. A mathematical model for balking is constructed by assigning a probability to balking dependent only on the state of the system; if an incoming call finds exactly j lines busy, then it realizes a connection with probability pj and balks with probability qj ( p j + (jj = 1). Thus if pj = 1 (j = 0, 1, · · ·) the system is one with an infinite number of lines and with no loss and no delay, the ideal for for any service, while if pj = 1 (j = 0, 1, m -- 1) and pj = 0 (j = m, m + 1, · · ·) the system is a loss system with m lines. This balking model is examined here for recurrent input and exponential distribution of holding times. More specifically, the call arrival times are taken as the instants n , r 2 , · · ·, r,, , · · ·, where the interarrival times 0n = r, l+ i -- r,t (n = 0, 1, · · ·; r0 = 0) are identically distributed, mutually independent, positive random variables with distribution function P{0,, ^ a:} = F(x). (1)