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Let the function $a(t),~ t \ge 0$, denotes the probability that there is an arrival in the current period (period $t$) given that no customers arrived in the previous $t-1$ periods. Assuming the inter arrival times to be independent and identically distributed, with $A$ as the generic inter arrival time. How to find the cumulative distribution of $A$?

Litun
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  • I find this confusing. Arrival times are often denoted by a continuous variable, t, but here you call it a "period", and speak of "t-1" periods between 0 and t-1 (when there are t of them, if t was an integer). If t was continuous, then A would be a random variable with an exponential distribution (given that the times between arrivals are iid), and the cumulative would be the integral of the exponential... – Dean May 07 '18 at 10:59
  • You are correct if we consider the continuous time queueing model, where the inter arrival times are represented by continuous random variables. But the present model is a discrete time queueing system, where the time axis is slotted into equal intervals, called periods. At most one customer can arrive in a period. Even if in the continuous case, I couldn't understand your claim that A will be exponentially distributed if inter arrival times are iid. I think A will be exponentially distributed only in case of Poisson arrivals. – Litun May 08 '18 at 04:40

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