# How do I determine the probability of a specific wait time in a queuing system?

I am working to model some queuing systems and running into a challenge answering a specific question.

I am looking to be able to graph the probability of different wait times. Example: probability a customer will wait between 0 and 1 minute, probability a customer will wait between 1 and 2 minutes, etc.

I have been unable to find an equation to answer this. My statistics isn't as good as it used to b, so if someone can help me work through an example it would be helpful.

Below is an example of single queue single server. Additional question on any answers, would I be able to apply to same to the results of a single queue, multi server analysis?

Example: Inputs
Time Unit hour
Arrival Rate (lambda) 5 customers/hour Service Rate (mu) 10 customers/hour

Intermediate Calculations
Average Time Between Arrivals 0.20 hour Average Service Time 0.10 hour

Performance Measures
Average Utilization (Rho) 50% Probability of empty system (P0) 50% Average Customers in System (L) 1 customers Average Customers in Line (Lq) 0.5 customers Average Time in System (W) 0.2 hour Average time in Queue (Wq) 0.10 hour

Probability of customer wait time

0-1 min ? 1-2 min ? 2-3 min ? etc

• I think I may have found it, but could use some validation that I am doing the right thing. – Peter Mar 3 '17 at 21:35
• I think I got it. – Peter Mar 3 '17 at 21:50
• p{Wq>t}=pe^-u(1-p)t – Peter Mar 3 '17 at 22:01
• Are there any assumptions on the distribution of the interarrival and service times aside from their mean? If not, only estimates can be given. – Math1000 Mar 3 '17 at 22:02
• Assumption is Poisson distribution for interarrival and exponential distribution for service. I think. – Peter Mar 3 '17 at 23:53