My question is below:

Consider a network of n queues with a Poisson arrival process of parameter t from outside the network, and independent exponentially distributed service times of parameters r1 to rn.

Customer that first arrived to the network initally join queue i with probability Pi (obviously there probabilities sum to one)

A customer completeing its service at queue i will head to queue j with probability Pij, or will leave the network with probability $$ P_{i, n+1} $$

Let K(t) denote the queue length vector for each of the queues at time t, and let $$ l = (k_{1},...,k_{2}) $$, k1 to kn is >= 0

be a particular value of this vector

Obtain p(k) = lim(t->inf) p(k,t) where p(k,t) = Prob[K(t) = k] in terms of the parameters previously defined, and prove your result in full detail

So I immediately tell that we are talking about the network going into a steady state.

To begin solving it I considered the small time interval delta t, where 1 packet may arrive, 0 arrive or many may arrive for each node. Each event has a probability. When we work through the maths it is possible to subtract from both sides of the equation

Ultimately I express $$ \frac{d}{dt}p(k,t) = all-possible-events $$

In steady state the rate of change will be 0, therefore $$ \frac{d}{dt}p(k,t) = 0 $$.

This is where I am stuck. Now what do I do? I know from other reading that the result will be p(k) = product form where the nodes behave independently of one another (a powerful result). But how do I find p(k) from $$ \frac{d}{dt}p(k,t) $$ to get p(k). I don't think I can just integrate, nor do I solve a differential equation. What should I do from here?

Let me know if you don't understand what I'm asking :)

I can do the proof afterwards



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