Queuing Model = mms, Arrival Rate `lambda=10` per 1 hr, Service Rate `mu=3` per 1 hr, Number of servers `s=4`

Solution:
Arrival Rate `lambda=10` per 1 hr and Service Rate `mu=3` per 1 hr (given)

Queuing Model : M/M/s

Arrival rate `lambda=10,` Service rate `mu=3,` Number of servers `s=4` (given)


1. Traffic Intensity
`rho=lambda/mu`

`=(10)/(3)`

`=3.33333333`


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+(3.33333333)^1/(1!)+(3.33333333)^2/(2!)+(3.33333333)^3/(3!) + (3.33333333)^4/(4!)*(4*3)/(4*3-10)]^(-1)`

`=[1+(3.33333333)/(1)+(11.11111111)/(2)+(37.03703704)/(6) + (123.45679012)/(24)*(12)/(2)]^(-1)`

`=[1+3.33333333+5.55555556+6.17283951 + 30.86419753]^(-1)`

`=[46.92592593]^(-1)`

`=0.02131018` or `0.02131018xx100=2.131018%`


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

`=((3.33333333)^4/(3!)*(10*3)/(4*3-10)^2)*0.02131018`

`=((123.45679012)/6*(30)/(2)^2)*0.02131018`

`=(154.32098765)*0.02131018`

`=3.28860826`


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

`=3.28860826/10`

`=0.32886083` hr or `0.32886083xx60=19.73164957` min

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

`=((3.33333333)^4/(3!)*(3)/(4*3-10)^2)*0.02131018`

`=((123.45679012)/6*(3)/(2)^2)*0.02131018`

`=(15.43209877)*0.02131018`

`=0.32886083` hr or `0.32886083xx60=19.73164957` min


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

`=3.28860826+3.33333333`

`=6.62194159`


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

`=6.62194159/10`

`=0.66219416` hr or `0.66219416xx60=39.73164957` min

Or
`W_s=W_q+1/mu`

`=0.32886083+1/3`

`=0.32886083+0.33333333`

`=0.66219416` hr or `0.66219416xx60=39.73164957` min


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

`=3.33333333/4`

`=0.83333333` or `0.83333333xx100=83.333333%`

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

`=(6.62194159-3.28860826)/(4)`

`=0.83333333` or `0.83333333xx100=83.333333%`


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>=4):}`

`P_n={(((3.33333333)^n)/(n!)*P_0, "for "0<=n<4),(((3.33333333)^n)/(4!*4^(n-4))*P_0, "for "n>=4):}`

`P_1=((3.33333333)^1)/(1!)*P_0=3.33333333/1*0.02131018=0.07103394`

`P_2=((3.33333333)^2)/(2!)*P_0=11.11111111/2*0.02131018=0.1183899`

`P_3=((3.33333333)^3)/(3!)*P_0=37.03703704/6*0.02131018=0.13154433`

`P_4=((3.33333333)^4)/(4!*4^(4-4))*P_0=123.45679012/(24*4^(0))*0.02131018=0.10962028`

`P_5=((3.33333333)^5)/(4!*4^(5-4))*P_0=411.52263374/(24*4^(1))*0.02131018=0.09135023`

`P_6=((3.33333333)^6)/(4!*4^(6-4))*P_0=1371.74211248/(24*4^(2))*0.02131018=0.07612519`

`P_7=((3.33333333)^7)/(4!*4^(7-4))*P_0=4572.47370828/(24*4^(3))*0.02131018=0.06343766`

`P_8=((3.33333333)^8)/(4!*4^(8-4))*P_0=15241.57902759/(24*4^(4))*0.02131018=0.05286472`

`P_9=((3.33333333)^9)/(4!*4^(9-4))*P_0=50805.26342529/(24*4^(5))*0.02131018=0.04405393`

`P_10=((3.33333333)^10)/(4!*4^(10-4))*P_0=169350.8780843/(24*4^(6))*0.02131018=0.03671161`

---