Poisson process: exponential inter arrival time vs exponential holding time

Inter arrival times in a Possion process is exponentially distributed. So I assume when I use the below command the ouputs follow that definition.

services= poissrnd(20,1,4)  %like for 4 time units. t=1,2,3,4

But I want a traffic generation where each traffic service also has a service duration which is exponentially distributed. I believe this is different from the exp inter-arrival time aspect.

So, how can I generate a duration for each service that is exp distributed so that I can depart it from the system once that duration is over. Do I need separate exp distribution? How can i connect the two then?

For e.g."Poisson process with an avg. arrival rate of λ requests per time-unit, and the lifetime of each request following negative exponential distribution with an average of 1/μ time units. So that the traffic load is λ/μ"

Answer 1

[Hornett]

To simulate a system where arrivals follow a Poisson process and service durations are exponentially distributed, you'll use MATLAB's exprnd function for both inter-arrival times and service durations, not poissrnd. Here's a brief guide:

Generating Inter-Arrival Times

For an average arrival rate λ:

lambda = 5; % Example average arrival rate
numServices = 4; % Number of services
interArrivalTimes = exprnd(1/lambda, 1, numServices);

Generating Service Durations

For an average service rate μ:

mu = 10; % Example service rate
serviceDurations = exprnd(1/mu, 1, numServices);

Connecting Both

Calculate arrival times by summing the inter-arrival times:

arrivalTimes = cumsum(interArrivalTimes);

This way, you have interArrivalTimes for when services enter the system and serviceDurations for how long each service lasts, modeling a system with traffic load characterized by λ/μ.

Follwing documentation might help you in your implementation

• exprnd: https://www.mathworks.com/help/stats/exprnd.html
• cumsum: https://www.mathworks.com/help/matlab/ref/cumsum.html

Hope this helps!