Question

1. (Queuing Model) A single public health worker is inoculating children against measles at a local clinic. An average of 8 children per hour is expected to arrive according to a Markovian process. He serves the children at a rate of 10 per hour. Assume that the service process is also approximately a Markovian process.

1. For the ‘inoculation system’, find:

1. the utilization of the health worker.
2. the average time spent by a child in the system.
3. the average number of children in the system.
4. the average number of children in line.
5. the average number of children in the system if another health worker is added to the system who has the same service rate as the existing one (assuming that there is one waiting line that is shared between the two health workers).

B. At what rate must children be served by the single public health worker so that the average time that a child spends in the system can be reduced by 50%?

Answer 1

Normal appearance rate, λ = 8 for every hr.

Normal help rate, μ = 10 for every hr.

a.

(I) Utilization = λ/μ = 8/10 = 80%

(ii) Average time spent in the framework, Ws = 1/(μ - λ) = 1/(10 - 8) = 0.5 hrs. = 30 minutes

(iii) Average number in the framework, Ls = Ws * λ = 0.50*8 = 4

(iv) Average number in the line, Lq = λ2/{μ.(μ - λ)} = (8^2)/(10*(10-8)) = 3.2

(v)

Number of servers, m = 2

Utilize the accompanying two equation to discover the idel server likelihood (P0) and normal number in the line (Lq).

— for 5.11> . 2.4(alu) (s – 1)!(s.1-2) 9

In this way,

P0 = 1/((1 + (8/10)^1) + (1/2)*(8/10)^2*(2*10)/(2*10-8)) = 0.4286

Lq = 8*10*(8/10)^2*0.4286/(1*(2*10-8)^2) = 0.15

In this way, normal number in the framework, Ls = Lq + λ/μ = 0.15+0.80 = 0.95

b.

The normal time in the framework (Ws) needs to lessen by half for example from 30 minutes to 15 minutes or 0.25 hrs,

In this way,

Ws = 1/(μ - λ) = 0.25

or then again, 1/(μ - 8) = 0.25

or then again, μ = 12

In this way, the administration rate must be 12 every hour.