n the next The density of the exponential is e-45 and integration of this infinitesimal time interval dt is Q dt, given it has not been completed gives the corresponding distribution function 1 - c-4's Suppose X is already. the random variable drawn from this distribution. Say that X, The Poisson process may be constructed from the exponential. "completes" when the time corresponding to X is reached starting Indeed, it is useful to regard the Poisson process as a sequence of from 0). Suppose seconds have elapsed and X has not completed. customers arriving at a queue with mutually independent and ident- The memoryless property is this: that the distribution of remaining ical exponential interarrival times. As such, the Poisson process time until X completes in no way depends on t and is given by the `inherits' the memoryless property from the exponential. This same exponential distribution as X. means that the probability an arrival takes place in any given 3/6 Queuing theory interval is independent of the history of the queue prior to that The conditioning event must have positive probability. In words, interval and in particular of the state of the queue immediately the probability of finishing up in a particular state conditional on the before it arrives. The parameter of the Poisson process is the same process visiting given states at given times depends only on the most rate parameter as the underlying exponential distribution. recent of those states stipulated. Thus to determine the probability In fact given a sequence of independent random variable, X1,X2 of future events one need only take the current state into account. with common exponential distribution rate parameter a, the Notice that the above probability, the transition probability fromjn.1 sequence of arrival times is determined by expression 3.3, starting to j, may also depend on the time, but if it depends only on the from time (). difference s - the process is called time homogeneous. If, in addition, any state can be attained from any other st