# Confidence interval of mean of queue length (M/M/1)

#### Knaapje

##### New Member
Goodday,

For a simulation course I had to simulate a single server queue with Poisson processes for arrival and departure (M/M/1). Now, this really was not the issue, the issue is where I personally want to add a kind of validity check, which naturally can only be done with statistics.

I want to construct a 99.5% confidence interval for the average queue length. The following information is known: total simulated 'time', list of amount of time per queue length during the duration (example: w_0 = 3 means that during the entire simulation the total amount of time the queue contained no customers is 3) and all arrival and departure times and the analytical value of the mean.

Now, I know the formula for the weighted sample mean and the weighted sample variance (where the mean is known), but I have no idea how to construct the confidence interval. Any help would be greatly appreciated.

#### rogojel

##### TS Contributor
hi,
a simple and pragmatic way would be to use bootstapping, as you already habe a simulation. Of course if you are interested in a formula then it is not helping.

regards
rogojel

#### Knaapje

##### New Member
First of all thanks for your reply.

Hm, I'm afraid I'm not familiar with that term, but from what I can gather, I'm supposed to get a large number of estimates for the variable I want my confidence interval for and treat that set as the distribution for my variable. Then 'as normal' I'd have to find the left bound such that (1-confidence)/2 of the estimates are on the left of it and my right bound analogously. Am I correct?

#### rogojel

##### TS Contributor
hi,
yes, exactly.Of course large can be very large if you have a simulation.
regards
rogojel