Queuing Model = mm1n, Arrival Rate `lambda=1/10` per 1 hr, Service Rate `mu=1/4` per 1 hr, Capacity `N=5`
Solution:
Arrival Rate `lambda=1/10` per 1 hr and Service Rate `mu=1/4` per 1 hr (given)
Queuing Model : M/M/1/N
Arrival rate `lambda=0.1,` Service rate `mu=0.25,` Capacity `N=5` (given)
1. Traffic Intensity
`rho=lambda/mu`
`=(0.1)/(0.25)`
`=0.4`
2. Probability of no customers in the system
`P_0=(1-rho)/(1-rho^(N+1))`
`=(1-0.4)/(1-(0.4)^(5+1))`
`=(0.6)/(1-(0.4)^6)`
`=(0.6)/(0.995904)`
`=0.60246771` or `0.60246771xx100=60.246771%`
3. Probability of N customers in the system
`P_N=rho^N*P_0`
`=(0.4)^5*0.60246771`
`=0.01024*0.60246771`
`=0.00616927`
4. Average number of customers in the system
`L_s=rho/(1-rho) - ((N+1)*rho^(N+1))/(1-rho^(N+1))`
`=0.4/(1-0.4) - ((5+1)*(0.4)^(5+1))/(1-(0.4)^(5+1))`
`=0.4/0.6 - (6*(0.4)^6)/(1-(0.4)^6)`
`=0.66666667 - (6*(0.004096))/(1-(0.004096))`
`=0.66666667 - (0.024576)/(0.995904)`
`=0.66666667 - 0.02467708`
`=0.64198959`
5. Effective Arrival rate
`lambda_e=lambda(1-P_N)`
`=0.1*(1-0.00616927)`
`=0.09938307`
6. Average number of customers in the queue
`L_q=L_s-(lambda_e)/(mu)=L_s-(lambda(1-P_N))/(mu)`
`=0.64198959-0.09938307/0.25`
`=0.2444573`
7. Average time spent in the system
`W_s=(L_s)/(lambda_e)=(L_s)/(lambda(1-P_N))`
`=(0.64198959)/(0.09938307)`
`=6.45974782` hr or `6.45974782xx60=387.58486906` min
8. Average Time spent in the queue
`W_q=(L_q)/(lambda_e)=(L_q)/(lambda(1-P_N))`
`=(0.2444573)/(0.09938307)`
`=2.45974782` hr or `2.45974782xx60=147.58486906` min
9. Utilization factor
`U=L_s-L_q`
`=0.64198959-0.2444573`
`=0.39753229` or `0.39753229xx100=39.753229%`
10. Probability that there are n customers in the system
`P_n=rho^n*P_0`
`P_n=(0.4)^n*P_0`
`P_1=(0.4)^1*P_0=0.4*0.60246771=0.24098708`
`P_2=(0.4)^2*P_0=0.16*0.60246771=0.09639483`
`P_3=(0.4)^3*P_0=0.064*0.60246771=0.03855793`
`P_4=(0.4)^4*P_0=0.0256*0.60246771=0.01542317`
`P_5=(0.4)^5*P_0=0.01024*0.60246771=0.00616927`
`P_6=(0.4)^6*P_0=0.004096*0.60246771=0.00246771`
`P_7=(0.4)^7*P_0=0.0016384*0.60246771=0.00098708`
`P_8=(0.4)^8*P_0=0.00065536*0.60246771=0.00039483`
`P_9=(0.4)^9*P_0=0.00026214*0.60246771=0.00015793`
`P_10=(0.4)^10*P_0=0.00010486*0.60246771=0.00006317`
This material is intended as a summary. Use your textbook for detail explanation.
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