Queuing Model = mms, Arrival Rate `lambda=1` per 5 hr, Service Rate `mu=1` per 1 hr, Number of servers `s=4`

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

Queuing Model : M/M/s

Arrival rate `lambda=0.2,` Service rate `mu=1,` Number of servers `s=4` (given)


1. Traffic Intensity
`rho=lambda/mu`

`=(0.2)/(1)`

`=0.2`


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+(0.2)^1/(1!)+(0.2)^2/(2!)+(0.2)^3/(3!) + (0.2)^4/(4!)*(4*1)/(4*1-0.2)]^(-1)`

`=[1+(0.2)/(1)+(0.04)/(2)+(0.008)/(6) + (0.0016)/(24)*(4)/(3.8)]^(-1)`

`=[1+0.2+0.02+0.00133333 + 0.00007018]^(-1)`

`=[1.22140351]^(-1)`

`=0.81873025` or `0.81873025xx100=81.873025%`


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

`=((0.2)^4/(3!)*(0.2*1)/(4*1-0.2)^2)*0.81873025`

`=((0.0016)/6*(0.2)/(3.8)^2)*0.81873025`

`=(0.00000369)*0.81873025`

`=0.00000302`


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

`=0.00000302/0.2`

`=0.00001512` hr or `0.00001512xx60=0.00090718` min

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

`=((0.2)^4/(3!)*(1)/(4*1-0.2)^2)*0.81873025`

`=((0.0016)/6*(1)/(3.8)^2)*0.81873025`

`=(0.00001847)*0.81873025`

`=0.00001512` hr or `0.00001512xx60=0.00090718` min


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

`=0.00000302+0.2`

`=0.20000302`


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

`=0.20000302/0.2`

`=1.00001512` hr or `1.00001512xx60=60.00090718` min

Or
`W_s=W_q+1/mu`

`=0.00001512+1/1`

`=0.00001512+1`

`=1.00001512` hr or `1.00001512xx60=60.00090718` min


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

`=0.2/4`

`=0.05` or `0.05xx100=5%`

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

`=(0.20000302-0.00000302)/(4)`

`=0.05` or `0.05xx100=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>=4):}`

`P_n={(((0.2)^n)/(n!)*P_0, "for "0<=n<4),(((0.2)^n)/(4!*4^(n-4))*P_0, "for "n>=4):}`

`P_1=((0.2)^1)/(1!)*P_0=0.2/1*0.81873025=0.16374605`

`P_2=((0.2)^2)/(2!)*P_0=0.04/2*0.81873025=0.0163746`

`P_3=((0.2)^3)/(3!)*P_0=0.008/6*0.81873025=0.00109164`

`P_4=((0.2)^4)/(4!*4^(4-4))*P_0=0.0016/(24*4^(0))*0.81873025=0.00005458`

`P_5=((0.2)^5)/(4!*4^(5-4))*P_0=0.00032/(24*4^(1))*0.81873025=0.00000273`

`P_6=((0.2)^6)/(4!*4^(6-4))*P_0=0.000064/(24*4^(2))*0.81873025=0.00000014`

`P_7=((0.2)^7)/(4!*4^(7-4))*P_0=0.0000128/(24*4^(3))*0.81873025=0.00000001`

`P_8=((0.2)^8)/(4!*4^(8-4))*P_0=0.00000256/(24*4^(4))*0.81873025=0`

`P_9=((0.2)^9)/(4!*4^(9-4))*P_0=0.00000051/(24*4^(5))*0.81873025=0`

`P_10=((0.2)^10)/(4!*4^(10-4))*P_0=0.0000001/(24*4^(6))*0.81873025=0`

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