I have this question I am struggling with:
Suppose that packet interarrival time and packet service times at a router follow an exponential probability density function and the router will be handling a mean arrival of 2500 packets per second.
Suppose that the router has 1MB of memory for its incoming queue and is processing 2700 packets per second on average. The average packet size is 1KB. The network administrator needs to tune the router to use some CPU time for other tasks like routing for network management. If the network administrator is comfortable with at most 80% of the queue memory space used (on average) then what percentage of CPU time can be used for these other tasks?
I am really not sure if I got this concept right, so here is my solution and my questions:
1MB memory
µ = 2700
Average packet size: 1KB = 1024 bytes
%80 memory space = 0.8 * 1MB = 838861 bytes
N = 819 (number of packets the queue can hold)
From this point I assumed the arrival is the same with part a, λ=2500. I also assumed that my CPU time is 2700 in total and all used if 1mb queue is %100 used.
My reasoning is that: what CPU time(out of 2700) would I need to maintain the N=819 packets in the queue with 2500 arrival time?
So I used N = λ / (µ - λ) and found µ=2503
Then I would have 2503/2700 * 100 = %93 which means I have %7 CPU time left for other stuff.
Is this how I am supposed to interpret and solve this question? I have a feeling that I am totally off the track or I am missing something. Thanks for your time!
The phrasing "at most 80% of the queue memory space used (on average)" strikes me as rather unclear. You might want to ask for clarification about exactly what that means, I will offer one interpretation here.
The model described is an M/M/1 queue with finite capacity with a state space like that shown below.
You have an arrival rate $\lambda = 2500$ packets per second and a potential service rate $\mu =2900$ packets per second. The M/M/1/$n$ model has storage space for $N$ jobs, so as 1MB = 1024 KB we set $n=1024$. For this model now we can compute the average number of jobs in the system. (I'll leave that for you to do.)
What the question asks is what value of $p$ can we choose (setting the service rate to be $\mu = 2900p$) such that the average number of customers in the system is 80% (i.e. 819). You need to use the calculation I left as an exercise above to compute the average number of jobs in the system as a function of $p$ and then choose the value of $p$ which sets the average number of jobs in the system to be 80%.
If this question is homework please tag it as such. The question feels rather unclear, perhaps you can double check your source?