Let us denote  as the total number of renewals that occur and then  with probability 1. It must be remembered that  is finite only when one of the inter arrival rates, i.e.,  is infinite for any

Thus


                 

                

Hence  goes to infinity as  also goes to infinity, but what is of more interest to us is the rate of increase of  with respect to , i.e., we are interested to find .

So first consider the random variable  and the significance it has for determining our above answer.  represents the time for the  event and since there are only  number of events that have occurred before time .  also represents the time for the last event, i.e.,  event prior to or at time . Thus one should clearly understand that  represents the time of the last renewal prior or at time , while  represents the time of the first renewal after time . Using a pictorial illustration we show this in Figure 4.5.