The average response time on a database system is 3 seconds

The average response time on a database system is 3 seconds. During a 1-minute observation interval, the idle time on the system was measured to be 10 seconds. Using an M/M/1 model for the system, determine the following: System utilization Average service time per query Number of queries completed during the observation interval3 Average number of jobs in the system Probability of number of jobs in the system being greater than 10 90-percentile response time 90-percentile waiting time

Solution

Given Data:

Average Response Time = 3 seconds
Observation interval = 1 minute = 60 seconds
Idle time = 10 seconds

Solution:

a. System Utilization is =5/6

b. Average service time per query E[s] = (1/)

We will be using E[r] = (1/) / (1-)

Hence (1/) = E[r] * (1-)

So E[s] = 1 / = E[r]*(1-) =
= 3 * (1-5/6)
= 3 * (1/6)
= 0.5s

c. Number of queries completed during the observation interval 3

x = 50/0.5 = 100

d. Average number of jobs in the system

E[n] = E[nq]+
= E[w]*+
=(E[r]-E[s])*(*)+
=(3-0.5)*(5/6*2)+5/6
= 5/2 * 5/3 + 5/6
= 25/6 + 5/6
= 30/6
= 5

e. Probability of number of jobs in the system being greater than 10

P[n>=10] = p10 = rho^11 = (5/6)^11 = 0.1346

f. 90-percentile response time

q-percentile of E[r] = E[r] * ln(100/(100-q))
So 90-percentile = E[r]*ln(10) = 6.9s

g. 90-percentile waiting time

E[wq] = E[r]-E[s] = 2.5s. q-percentile of E[wq]
= (E[wq]/rho) * ln(100rho/(100-q))
= 6.36