Queuing Model = mm1n, Arrival Rate `lambda=6` per 1 hr, Service Rate `mu=7` per 1 hr, Capacity `N=3`
Solution:
Arrival Rate `lambda=6` per 1 hr and Service Rate `mu=7` per 1 hr (given)
Queuing Model : M/M/1/N
Arrival rate `lambda=6,` Service rate `mu=7,` Capacity `N=3` (given)
1. Traffic Intensity
`rho=lambda/mu`
`=(6)/(7)`
`=0.85714286`
2. Probability of no customers in the system
`P_0=(1-rho)/(1-rho^(N+1))`
`=(1-0.85714286)/(1-(0.85714286)^(3+1))`
`=(0.14285714)/(1-(0.85714286)^4)`
`=(0.14285714)/(0.46022491)`
`=0.31040724` or `0.31040724xx100=31.040724%`
3. Probability of N customers in the system
`P_N=rho^N*P_0`
`=(0.85714286)^3*0.31040724`
`=0.62973761*0.31040724`
`=0.19547511`
4. Average number of customers in the system
`L_s=rho/(1-rho) - ((N+1)*rho^(N+1))/(1-rho^(N+1))`
`=0.85714286/(1-0.85714286) - ((3+1)*(0.85714286)^(3+1))/(1-(0.85714286)^(3+1))`
`=0.85714286/0.14285714 - (4*(0.85714286)^4)/(1-(0.85714286)^4)`
`=6 - (4*(0.53977509))/(1-(0.53977509))`
`=6 - (2.15910037)/(0.46022490628904633)`
`=6 - 4.69140271`
`=1.30859729`
5. Effective Arrival rate
`lambda_e=lambda(1-P_N)`
`=6*(1-0.19547511)`
`=4.82714932`
6. Average number of customers in the queue
`L_q=L_s-(lambda_e)/(mu)=L_s-(lambda(1-P_N))/(mu)`
`=1.30859729-4.82714932/7`
`=0.61900452`
7. Average time spent in the system
`W_s=(L_s)/(lambda_e)=(L_s)/(lambda(1-P_N))`
`=(1.30859729)/(4.82714932)`
`=0.27109111` hr or `0.27109111xx60=16.26546682` min
8. Average Time spent in the queue
`W_q=(L_q)/(lambda_e)=(L_q)/(lambda(1-P_N))`
`=(0.61900452)/(4.82714932)`
`=0.12823397` hr or `0.12823397xx60=7.69403825` min
9. Utilization factor
`U=L_s-L_q`
`=1.30859729-0.61900452`
`=0.68959276` or `0.68959276xx100=68.959276%`
10. Probability that there are n customers in the system
`P_n=rho^n*P_0`
`P_n=(0.85714286)^n*P_0`
`P_1=(0.85714286)^1*P_0=0.85714286*0.31040724=0.26606335`
`P_2=(0.85714286)^2*P_0=0.73469388*0.31040724=0.2280543`
`P_3=(0.85714286)^3*P_0=0.62973761*0.31040724=0.19547511`
`P_4=(0.85714286)^4*P_0=0.53977509*0.31040724=0.1675501`
`P_5=(0.85714286)^5*P_0=0.46266437*0.31040724=0.14361437`
`P_6=(0.85714286)^6*P_0=0.39656946*0.31040724=0.12309803`
`P_7=(0.85714286)^7*P_0=0.33991668*0.31040724=0.1055126`
`P_8=(0.85714286)^8*P_0=0.29135715*0.31040724=0.09043937`
`P_9=(0.85714286)^9*P_0=0.2497347*0.31040724=0.07751946`
`P_10=(0.85714286)^10*P_0=0.21405832*0.31040724=0.06644525`
This material is intended as a summary. Use your textbook for detail explanation.
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