c. The arrival of large jobs at a computer centre forms a Poisson process with rate 2 per hour. The service time of such jobs are exponentially distributed with mean 20 minutes. Only 4 large jobs can be accommodated in the system at a time.
i. Determine the probability that a large job will be turned away because of lack of storage.
ii. What is the probability that the arriving customer has to wait?
To answer these questions, we can use the concepts of queueing theory. Specifically, we'll need to apply the M/M/c queueing model.
In the M/M/c model, "M" refers to a Poisson arrival process, "M" refers to the exponential service time distribution, and "c" refers to the number of servers or channels in the system.
i. To determine the probability that a large job will be turned away because of lack of storage, we need to calculate the system's steady-state probability that there are already 4 large jobs in the system.
1. First, let's calculate the traffic intensity, ρ, which is the ratio of the arrival rate to the service rate per server:
ρ = (arrival rate) / (service rate per server)
ρ = 2 per hour / (1 job per 20 minutes) * (60 minutes / 1 hour)
ρ = 2 / (1/3)
ρ = 2 * 3
ρ = 6
2. Next, let's calculate the utilization factor, U, which is the ratio of ρ to the number of servers, c:
U = ρ / c
U = 6 / 4
U = 1.5
3. Now, using the formula for the steady-state probability of having all servers occupied in an M/M/c queue:
P(all servers occupied) = (U^c / c!) * (1 / (1 - U)) * (1 / Σ(i=0 to c-1) (U^i / i!))
P(all servers occupied) = (1.5^4 / 4!) * (1 / (1 - 1.5)) * (1 / (1 + 1.5 + 1.5^2 + 1.5^3))
P(all servers occupied) = (0.421875 / 24) * (-2/3) * (1 / (1 + 1.5 + 2.25 + 3.375))
(Note: The negative sign in -2/3 is due to the denominator of 1 - U, where U > 1)
P(all servers occupied) = 0.017579
Therefore, the probability that a large job will be turned away because of lack of storage is approximately 0.017579, or about 1.76%.
ii. To calculate the probability that the arriving customer has to wait, we need to consider the probability that all servers are occupied when a job arrives.
1. The probability that the system is full (all servers occupied) can be calculated by using the formula we derived in part i:
P(all servers occupied) = 0.017579
2. The probability that the arriving customer has to wait is therefore the complement of the probability that the system is full:
P(waiting) = 1 - P(all servers occupied)
P(waiting) ≈ 1 - 0.017579
P(waiting) ≈ 0.982421
Therefore, the probability that the arriving customer has to wait is approximately 0.982421, or about 98.24%.