Stochastic processes: infinite server queue with batch poisson arrivals

[mkln]

Hi everyone, I am trying to solve this problem but I am stuck with doubts. Here are my ideas.

Homework Statement

Busloads of customers arrive at an infinite server queue at a Poisson
rate λ
Let G denote the service distribution. A bus contains j customers
with probability aj = 1, . Let X(t) denote the number of customers
that have been served by time t
(a) E(X(t)) = ?
(b) Is X(t) Poisson distributed?

Homework Equations

Basically the scenario is an infinite server queue with poisson independent batch arrivals

The Attempt at a Solution

In class we considered the infinite server queue with poisson independent arrivals.
We also mentioned compound poisson processes.
This problem merges the two.

I use Xt as the number of customers that have been served by time t, then:

Xt = Yt - It

where Yt is the total number of customers that have arrived up to time t,
and It is the number of customers that are still being served at time t.

Yt is a compound poisson process and we have that its expected value is:
E(Yt) = E(Nt) E(B1)
where {Nt} is the Poisson process describing the arrival of the buses, and E(B1) is the expected number of customers in a generic bus.
Here we can find $E(B1)=\sum{ja_j}$

My problem is to figure out E(It).
We did the same thing in class when the customers arrived independently (not in batches).
The professor told me as a hint to condition on Nt=n and take E(E(Xt|Nt=n)).
I still have to figure out why I should use that. Unless he meant something else, Nt is the number of buses, and it being equal to n doesn't tell me how many customers have arrived.
In the non-batch arrival case, I know that conditioning on Nt=n makes the arrival times uniformly distributed. But how can I say the same if I have customers arriving in batches?


Should I try conditioning on the total number of customers arrived? something like
E( E( Xt | Yt = y )) ?
And if I do this, can arrival times still be considered as being distributed like a Uniform(0,t) ?

I looked for some info online and I actually found some papers on this infinite server queue with batch arrivals. But they are all too complicated. First I don't understand most of them, second this problem is supposed to be very easy.

I hope you have some insights to share! thanks!

 

[mighty2000]

Hi there, it seems like you have a good understanding of the problem so far. Conditioning on the total number of customers arrived (Yt) is the right approach. This is because, as you mentioned, Nt is the number of buses and does not directly relate to the number of customers that have arrived. By conditioning on Yt, you are taking into account the number of customers that have arrived, regardless of how many buses they arrived on.

As for the distribution of arrival times, it is still possible to consider them as being distributed like a Uniform(0,t). This is because, although the customers may arrive in batches, each individual customer within the batch still arrives at a random time within the batch's arrival time. Therefore, the overall arrival times can still be modeled as a Uniform(0,t) distribution.

I hope this helps and good luck with your solution!