By Frank A. Haight (Eds.)
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4 RANDOM ARRIVALS 47 Kâ(1) = p. (3) and mean value These quantities can be used to evaluate the probability of queue length changes between successive regeneration points. = Prob (state j at a regeneration point I state i at preceding regeneration point). Then, it is easy to see that Po, = k, p21=k1-z+1, p6, = 0 in all j-i+l>O, i>O (4) other cases. Moreover, the one stage transition probabilities pZ1must be related to the equilibrium probabilities p , through the formulas since each state j must have been preceded by some state i and the appropriate transition from i to i.
1. Wiley, New York, 1950. 3. Munroe, M. â McGraw-Hill, New York, 1951. 4. â Wiley, New York, 1958. 1 Stochastic Processes Suppose a random variable X with density f ( x ) is a function of a real variable t ; we think of X ( t ) as the value of X at time t , and of f ( x ; t ) as the density at time t. All the other quantities associated with X will be functions of t : mean, m(t); variance, v ( t ) ; Laplace transform, n ( s ; t ) ; etc. Then X ( t ) is called a stochastic pYocess with cumulative distribution function < x) F ( x ;t ) = Prob ( X ( t ) in the continuous case and P,(t) = Frob < ( N ( t ) n) in the discrete case.
The cumulative function corresponding to Fig. 3 is shown in Fig. 4. 4. A cumulative distribution for the mixed case. Cumulative functions are valuable theoretically, but for particular distributions they may be very cumbersome. GEOMETRIC DISTRIBUTION pn=cn (l-p)pi= +=0 POISSON DISTRIBUTION which has no simple form. l-pP"+l. (3) 18 PROBABILITY AND STATISTICS BINOMIAL DISTRIBUTION p,=z': 2=0 (Y),i(l-P)"-i (5) which has no simple form. TYPEI11 DISTRIBUTION Although this function has no simple form, it occurs often enough in applied mathematics to deserve a special name and symbol.
