6711-09-Notes-Discrete-Renewal

6711-09-Notes-Discrete-Renewal - Copyright c 2009 by Karl...

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Copyright c ± 2009 by Karl Sigman 1 Discrete-time renewal processes Imagine busloads of passengers, where the i th bus contains H i passengers, and the { H i } are iid with pmf p ( k ) = P ( H = k ) , k 1 . (1) Imagine further that the seats on a bus are labeled 1 , 2 ,...,H , from the back of the bus to the front in one long line. A passenger is said to be in the j th position on their bus if they are in the seat labeled j . We shall assume the expected bus size is positive, 0 < E ( H ) < , to avoid trivialities. (We also allow the bus to be of unlimited size, if necessary.) If we “randomly select a passenger way out among all passengers among all buses”, then it is of intrinsic interest to determine such quantities as 1. The distribution (and mean) of the position of the passenger in question. 2. The distribution (and mean) of the bus size containing the passenger in question. 3. The distribution (and mean) of the number of passengers in front of (or in back of) this passenger on the bus. A little thought reveals that to determine the above quantities (and to make this framework precise), we simply need to construct a discrete-time renewal process in which the “interarrival” times are the H i , and N ( n ) def = the number of bus arrivals by time n . The “arrival times” are

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6711-09-Notes-Discrete-Renewal - Copyright c 2009 by Karl...

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