Problem Detail: In a queuing system (M/M/1) with a finite packet capacity $z$, how do you determine the probability of packet loss if we assume that packets are dropped when the system is full? Packets arrive with a rate $lambda$ and are served at rate $mu$ (using the Poisson distribution). My attempt at a solution: Probability of packet loss = Probability that the system has exactly $n$ packets (i.e., it is at its capacity)? I also have a formula from my notes where probability of packet loss is approximated at $(1 – frac{lambda}{mu})(frac{lambda}{mu})^z$ (so this is most likely correct but can someone please explain why?). I’m not sure how to calculate the rate at which packets are dropped. I already know the probabilities $P(n)$ that there are $n$ packets in the system where $n = 0, 1, …, z$ packets.
Asked By : 0kB
Answered By : arcticriki
If you study M/M/1 queuing systems you’ll probably know that such a queuing system can be seen as a markovian chain with birth rate λ and death rate $mu$; that is to say that you change your state (and in an M/M/1 the state is the number of clients in the system) by unitary steps: $-1$ with rate $mu$ and $+1$ with rate $lambda$. It is quite hard and long to explain in a thread like this and for this reason I suggest you to take a look at the markovian chains theory to understand why it works like this. Anyway, I’ll try to give you an idea: you can set a basic fluxus equation to describe the system: $$P[x(t)=k] mu = P[x(t)=k-1] lambda$$ and analyzing the transition probabilities you will find: $$ P[x(t+h)=k+1 | x(t)=k] = lambda h + o(h) P[x(t+h)=k-1 | x(t)=k] = mu h + o(h) P[x(t+h)=k | x(t)=k] = 1-lambda h-mu h + o(h) P[x(t+h)=k+j | x(t)=k] = h + o(h)$$ with $h$ defined as very little piece of time. So, defined $ρ=frac{lambda}{mu}$ as the load factor of the system, you will find that (recalling the total probability theorem): $$ P[x(t)=k] = (1-ρ) ρ^k $$ If you take a look at this result, you will recognize the answer to your question: if the system is a blocking system with blocking capacity $z$, the probability of packet loss is the following: $$P[z,text{clients in the system}] = P[k,text{clients in the system}] = P[x(t)=k] = (1-ρ) ρ^k$$ Is it ok? 🙂
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Question Source : http://cs.stackexchange.com/questions/37975
