Within a queue with capacity = K and exponential interarrival times, death rate is μ and birth rate λ.
A packet is discarded when the queue is full.
When the source is active there's a probability p1 > 0 that a packet is discarded at arrival. Moreover there's a probability p2 > 0 that all packets are discarded at arrival.
(so, a packet enters the queue with P=(1-p1-p2 | source is active))
The source that sends the packets with λ (birth rate) is idle for exponentially distributed times with rate ℬ (and active with rate ℒ)
Model the process with a Markov chain.
This is an extension to this excercise: variable death / birth rate in stochastic process
My thoughts follow:
I suppose, without considering idle periods, arrival times are poissonian with a birth rate of λ(1-p1-p2)
Now consider the source idle time. It is a poissonian process itself with only two states (active and idle). So we have a "random" selection of poissonian arrivals with probability P$\_a$ = P(source is active) which we know has an exponential distribution with parameter ℒ. Birth rate becomes $λ(1-p1-p2)ℒ\Delta te^{-ℒ\Delta t}$ (not really sure about this).
Death rate changes according to probability p2: - $S_{n,n-1} = μ(1-p2)$ - $S_{n,0} = \frac{n p2}{\Delta t}$
So having poissionian arrivals but non-poissonian departures we are in a M/G/1/K queue with transition rates when the source is active (EDITED, see comments)
I've drawn a state transition diagram welcoming @D.Thomine suggestions in comments.
(not sure about it yet, I suspect I've to substract rate ℬ...)