Is the number of customers in the system. States of the system, that are necessary to compute the required quantitative parameters. Because both arrival and service are Poisson processes, it is possible to find probabilities of various M ), unlimited FIFO (or not specified queue), and unlimited customer L ), one exponential server (Service rate The M/M/1 system is made of a Poisson arrival (Arrival rate Later we shall see that in fact r must not be even equal to one. To find its distribution, let’s express the Distribution function F( x):į( x) = P[interval 1 the queue will grow permanently. That’s why the l is called Arrival rate.Īnother random variable associated with the Poisson process is the random interval between two adjacent arrivals. (5) gives the interpretation of the constant l, that is the average number of arrivals per time unit. Let E be the mean (average) value, Var the variance, and Std the standard deviation of the random variable x: Having the probabilities of the random values - (4), it is possible to find the usual parameters of the random variable N t. Number of arrivals N t during some time interval t is a discrete random variable associated with the Poisson process. Or generally (can be proved by mathematical induction):īecause of the assumption 3) the formula (4) holds for any interval ( s, s+t), in other words the probability of n arrivals during some time interval depends only on the length of this time interval (not on the starting time of the interval). (prove by computing the derivative and inserting to (3) with p 0( t) ) Using p 0( t) it is possible to compute p 1( t): (to prove it, compute the derivative of p 0( t) and compare with (3) ) The equations (1) may be written in this way:īecause of small h the terms at the left sides of (2) may be considered as derivatives:Įquations (3) represent a set of differential equations. ( n arrivals by t, no more arrival or n-1 arrivals by t, one more arrival) Using the above assumptions 1) and 2), it is possible to express the probability The Poisson process satisfies the following assumptions, where P means "theġ) P = l h, where l is a constant.Ģ) P ® 0.ģ) The above probabilities do not depend on t ("no memory" property = time independence = stationarity). Next two chapters summarize the basic properties of the Poisson processĪnd give derivation of the M/M/1 theoretical model. Nevertheless the M/M/1 model shows clearly the basic ideas and methods (the worst assumption is the exponential distribution of service duration - hardly satisfiedīy real servers). Note that these assumptions are very strong, not satisfied for practical systems Server, FIFO (or not specified) queue of unlimited capacity and unlimited customer population. The M/M/1 system is made of a Poisson arrival, one exponential (Poisson)
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