Question

less than or equal to 0.6ms (that is 600 usec). (2 poinS Problem 2: Queuing Theory 110 points] A computer system, A, generates packets destined to another Computer System B thru a router R. We are interested at the traffic analysis at the router R. The router receives packets as input, processes them and sends packets to Computer B. Packets are generated by Computer A at a rate of 85 packets/sec following a Poisson process Packets takes 6-8 milliseconds, in average, to be processed and sent on line to computer B (all delays included). The packet processing time (i.e., service time per packet) follows an exponential distribution with a given mean related to the packet length. Answer the following questions. Characterize this system, at the router in terms of what type of queuing system it is (according to Kendall's notation), and find the arrival rate of packets ( λ ), the 2.

application in broad band communications

Answer 1

Given, the rate of arrival of packets $\lambda$ = 85 packets/sec.

Time required by each packet is 8 millisecs

ie $\mu$ = 1/8ms

$\mu$= 125 packets/ sec

Therefor traffic intensity  $\rho = \lambda /\mu$

  $\rho$ = 85/125 = 0.68

The given system is stable since the departure rate is greater than arrival rate. $\mu> \lambda$

3. The estimated waiting time of packets in the system $W = 1/(\mu-\lambda)$ so

The estimated waiting time in the queue $Wq = \lambda / \mu (\mu -\lambda) = 85/(125*(125-85))= 0.017 = 17 ms$

4.The estimated number of packets in the queue$L = \lambda ^{2}/\mu(\mu-\lambda)= 85*85/125(125-85)=1.445 packets$

The estimated number of packets in the system $L = \lambda/\mu-\lambda=2.125 packets$

5. The minimum packet processing time which will make system unstable is when $\lambda = \mu$   $85 = 1/\theta ie \theta = 1/85 = 0.0117 = 11.7 ms$

6. The system is blocking now, the probability of 40 or more packets in the system is $P = \rho ^{k}$ this is the blocking probability of the system.