# Average response/waiting time for aggregated tasks with Poisson arrival

Suppose there is a specific computation task with Poisson arrival rate $$\lambda$$ that could be aggregated in a way that when a task arrives and triggers a computation which lasts for $$D$$ seconds, if other tasks of the same type arrive while its still in process, they will be aggregated with the first one. In other words, They all receive the response at the same time which is $$D$$ seconds after the first task arrival. I want to find the average response time.

It's been a while that I've been playing with exponential and gamma distributions to find this thing out, but what I've been getting never matches the simulation results that come from omnet++:

-$$D=30$$, $$\lambda=0.2$$ >> 17.14

-$$D=3$$, $$\lambda=2$$ >> 1.71

-$$D=3$$, $$\lambda=1$$ >> 1.87

-$$D=3$$, $$\lambda=0.2$$ >> 2.43

-$$D=3$$, $$\lambda=10$$ >> 1.55

-$$D=3$$, $$\lambda=100$$ >> 1.504788

-$$D=30$$, $$\lambda=10$$ >> 15.04788

Two things are clear from these results:

1. $$\lambda$$ decreases the result
2. The result is heavily dependent on $$\lambda*D$$ (see first and second for example)

Does anyone have any suggestions?

UPDATE:

user "Did" has posted a response for a similar problem here: Link

He considers the mean waiting time of every other passenger except the first one to be $$t/2$$ or $$D/2$$ for my problem. I'm not sure where it came from but it seems that it considers $$\lambda->\infty$$ which makes sense when I compare it to my simulations results that become closer with bigger $$\lambda$$.

>>My focus is on small $$\lambda$$s.<<

This is what he suggested:

$$w_t=E\left(\frac{t+\frac12tN_t}{1+N_t}\right)=\frac12t\left(1+E\left(\frac1{1+N_t}\right)\right)=\frac12t\left(1+\frac{1-e^{-\lambda t}}{\lambda t}\right)$$