Question

105.      Patients arrive at your Express Care clinic in the form of a Poisson distribution at a mean rate of 5.8 per hour. You have 3 providers per shift, and the service rate of seeing patients is exponentially distributed with a mean service rate of 2.3 patients per hour. Leadership has stated that patients should wait no more than 30 minutes for service. Answer the following questions: A.    What is the probability that there are no patients in the system? B. what is average number of patient waiting for service? c. what is average waiting time in mins. to see a provider? d.what is utilization of providers?

Answer 1

Arrival rate = 5.8 per hour

λ = 1/(5.8/60) =10.34 minutes

Service rate = 2.3 patients per hour

µ = 1/(2.3/60) = 26.09

c: No. of servers =3

ρ = λ/cµ = 10.34/(3*26.09) =0.13

a) What is the probability that there are no patients in the system?

P0 = 1/[(((1*0.13)^0)/0!) + (((1*0.13)^1)/1!) + (((2*0.13)^2)/2!) + ((3*0.13)^3)/(3!*(1-0.13))]

= 1/[1 + 0.13 + 0.0338 + 0.0113] =0.85

b) What is average number of patient waiting for service?

Lq = (0.85*((10.34/26.09)^3)*0.13)/(3!*(1-0.13)^2) =0.0015

c) What is average waiting time in mins?

Wq = Lq/λ = 0.0015/10.34 = 0.000146

d) What is utilization of providers?

ρ = λ/cµ = 10.34/(3*26.09) =0.13