Queuing Model = mms, Arrival Rate `lambda=15` per 1 hr, Service Rate `mu=1` per 5 min, Number of servers `s=2`

Solution:
Arrival Rate `lambda=15` per 1 hr and Service Rate `mu=1` per 5 min (given)

So, Arrival Rate `lambda=15` per hr and Service Rate `mu=0.2xx60=12` per hr

Queuing Model : M/M/s

Arrival rate `lambda=15,` Service rate `mu=12,` Number of servers `s=2` (given)


1. Traffic Intensity
`rho=lambda/mu`

`=(15)/(12)`

`=1.25`


2. Probability of no customers in the system
`P_0=[sum_{n=0}^(s-1) (rho^n)/(n!) + (rho^s)/(s!)*(s mu)/(s mu-lambda)]^(-1)`

`=[1+(1.25)^1/(1!) + (1.25)^2/(2!)*(2*12)/(2*12-15)]^(-1)`

`=[1+(1.25)/(1) + (1.5625)/(2)*(24)/(9)]^(-1)`

`=[1+1.25 + 2.08333333]^(-1)`

`=[4.33333333]^(-1)`

`=0.23076923` or `0.23076923xx100=23.076923%`


3. Average number of customers in the queue
`L_q=((rho^s)/((s-1)!)*(lambda mu)/(s mu-lambda)^2)*P_0`

`=((1.25)^2/(1!)*(15*12)/(2*12-15)^2)*0.23076923`

`=((1.5625)/1*(180)/(9)^2)*0.23076923`

`=(3.47222222)*0.23076923`

`=0.80128205`


4. Average Time spent in the queue
`W_q=L_q/lambda`

`=0.80128205/15`

`=0.0534188` hr or `0.0534188xx60=3.20512821` min

Or
`W_q=((rho^s)/((s-1)!)*(mu)/(s mu-lambda)^2) * P_0`

`=((1.25)^2/(1!)*(12)/(2*12-15)^2)*0.23076923`

`=((1.5625)/1*(12)/(9)^2)*0.23076923`

`=(0.23148148)*0.23076923`

`=0.0534188` hr or `0.0534188xx60=3.20512821` min


5. Average number of customers in the system
`L_s=L_q+rho`

`=0.80128205+1.25`

`=2.05128205`


6. Average Time spent in the queue
`W_s=L_s/lambda`

`=2.05128205/15`

`=0.13675214` hr or `0.13675214xx60=8.20512821` min

Or
`W_s=W_q+1/mu`

`=0.0534188+1/12`

`=0.0534188+0.08333333`

`=0.13675214` hr or `0.13675214xx60=8.20512821` min


7. Utilization factor
`U=rho/s=(lambda)/(s mu)`

`=1.25/2`

`=0.625` or `0.625xx100=62.5%`

Or
`U=(L_s-L_q)/s`

`=(2.05128205-0.80128205)/(2)`

`=0.625` or `0.625xx100=62.5%`


8. Probability that there are n customers in the system
`P_n={((rho^n)/(n!)*P_0, "for "0<=n< s),((rho^n)/(s!*s^(n-s))*P_0, "for "n>=2):}`

`P_n={(((1.25)^n)/(n!)*P_0, "for "0<=n<2),(((1.25)^n)/(2!*2^(n-2))*P_0, "for "n>=2):}`

`P_1=((1.25)^1)/(1!)*P_0=1.25/1*0.23076923=0.28846154`

`P_2=((1.25)^2)/(2!*2^(2-2))*P_0=1.5625/(2*2^(0))*0.23076923=0.18028846`

`P_3=((1.25)^3)/(2!*2^(3-2))*P_0=1.953125/(2*2^(1))*0.23076923=0.11268029`

`P_4=((1.25)^4)/(2!*2^(4-2))*P_0=2.44140625/(2*2^(2))*0.23076923=0.07042518`

`P_5=((1.25)^5)/(2!*2^(5-2))*P_0=3.05175781/(2*2^(3))*0.23076923=0.04401574`

`P_6=((1.25)^6)/(2!*2^(6-2))*P_0=3.81469727/(2*2^(4))*0.23076923=0.02750984`

`P_7=((1.25)^7)/(2!*2^(7-2))*P_0=4.76837158/(2*2^(5))*0.23076923=0.01719365`

`P_8=((1.25)^8)/(2!*2^(8-2))*P_0=5.96046448/(2*2^(6))*0.23076923=0.01074603`

`P_9=((1.25)^9)/(2!*2^(9-2))*P_0=7.4505806/(2*2^(7))*0.23076923=0.00671627`

`P_10=((1.25)^10)/(2!*2^(10-2))*P_0=9.31322575/(2*2^(8))*0.23076923=0.00419767`

---