Fundamentals of Probability and Statistics for Engineers

Some Important Discrete Distributions 175No arrivalNo arrival0 t t + ∆tFigure 6.2 Interval [0, t ‡ t)It follows from Equations (6.34) and (6.36) thatp 0 …t; t ‡ t† ˆ1In order to determine probability mass function p k (0, t) based on theassumptions stated above, let us first consider p 0 (0, t). Figure 6.2 shows twononoverlapping intervals, [0, t) and [t, t ‡ t). In order that there are noarrivals in the total interval [0, t ‡ t), we must have no arrivals in bothsubintervals. Owing to the independence of arrivals in nonoverlapping intervals,we thus can writeUpon letting t ! 0, we obtain the differential equationIts solution satisfying the initial condition p 0 (0, 0) ˆ 1 isX 1kˆ1p k …t; t ‡ t†ˆ 1 t ‡ o…t†: …6:37†p 0 …0; t ‡ t† ˆp 0 …0; t†p 0 …t; t ‡ t†ˆ p 0 …0; t†‰1 t ‡ o…t†Š: …6:38†Rearranging Equation (6.38) and dividing both sides by t gives p 0 …0; t ‡ t† p 0 …0; t†o…t†ˆ pt0 …0; t† :tdp 0 …0; t†dtˆ p 0 …0; t†: …6:39†p 0 …0; t† ˆe t :…6:40†The determination of p 1 (0,t) is similar. We first observe that one arrival in[0, t ‡ t) can be accomplished only by having no arrival in subinterval [0,t)and one arrival in [t, t ‡ t), or one arrival in [0,t) and no arrival in [t, t ‡ t).Hence we havep 1 …0; t ‡ t† ˆp 0 …0; t†p 1 …t; t ‡ t†‡p 1 …0; t†p 0 …t; t ‡ t†:…6:41†TLFeBOOK