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I am trying to understand about two distributions:

The system is customers are coming in and they are being serviced. There is a single server, and therefore, when a customer comes in, if server is busy, customer waits in the queue. Inter-arrival times of customers are exponentially distributed, and service times are also exponentially distributed.

M is the random variable denoting the number of customers seen by the arriving customer and

D is the random variable denoting the system time (wait time + service time).

Questions:

1) M should be discrete r.v. according to my understanding but when I draw the histogram for the values I get with my simulation: enter image description here However, this looks like pmf for exponential, and exponential distribution is continuous. What am I missing?

What would be the range of M if lambda=0.75 (arrival rate), M=1 (service rate)?

2) D should be continuous r.v. according to my understanding. And histogram gives this:

enter image description here

This also looks like pmf of exponential, which should be fine if D is continuous.

What would be the range of D, how can we know if we did not draw the histogram? I mean theoretically, what would be the way to figure out?

Thanks!

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