Queuing Model = mm1n, Arrival Rate `lambda=25` per 1 hr, Service Rate `mu=40` per 1 hr, Capacity `N=12`
Solution:
Arrival Rate `lambda=25` per 1 hr and Service Rate `mu=40` per 1 hr (given)
Queuing Model : M/M/1/N
Arrival rate `lambda=25,` Service rate `mu=40,` Capacity `N=12` (given)
1. Traffic Intensity
`rho=lambda/mu`
`=(25)/(40)`
`=0.625`
2. Probability of no customers in the system
`P_0=(1-rho)/(1-rho^(N+1))`
`=(1-0.625)/(1-(0.625)^(12+1))`
`=(0.375)/(1-(0.625)^13)`
`=(0.375)/(0.99777955)`
`=0.37583452` or `0.37583452xx100=37.583452%`
3. Probability of N customers in the system
`P_N=rho^N*P_0`
`=(0.625)^12*0.37583452`
`=0.00355271*0.37583452`
`=0.00133523`
4. Average number of customers in the system
`L_s=rho/(1-rho) - ((N+1)*rho^(N+1))/(1-rho^(N+1))`
`=0.625/(1-0.625) - ((12+1)*(0.625)^(12+1))/(1-(0.625)^(12+1))`
`=0.625/0.375 - (13*(0.625)^13)/(1-(0.625)^13)`
`=1.66666667 - (13*(0.00222045))/(1-(0.00222045))`
`=1.66666667 - (0.0288658)/(0.99777955395074969)`
`=1.66666667 - 0.02893004`
`=1.63773663`
5. Effective Arrival rate
`lambda_e=lambda(1-P_N)`
`=25*(1-0.00133523)`
`=24.96661919`
6. Average number of customers in the queue
`L_q=L_s-(lambda_e)/(mu)=L_s-(lambda(1-P_N))/(mu)`
`=1.63773663-24.96661919/40`
`=1.01357115`
7. Average time spent in the system
`W_s=(L_s)/(lambda_e)=(L_s)/(lambda(1-P_N))`
`=(1.63773663)/(24.96661919)`
`=0.06559705` hr or `0.06559705xx60=3.93582315` min
8. Average Time spent in the queue
`W_q=(L_q)/(lambda_e)=(L_q)/(lambda(1-P_N))`
`=(1.01357115)/(24.96661919)`
`=0.04059705` hr or `0.04059705xx60=2.43582315` min
9. Utilization factor
`U=L_s-L_q`
`=1.63773663-1.01357115`
`=0.62416548` or `0.62416548xx100=62.416548%`
10. Probability that there are n customers in the system
`P_n=rho^n*P_0`
`P_n=(0.625)^n*P_0`
`P_1=(0.625)^1*P_0=0.625*0.37583452=0.23489658`
`P_2=(0.625)^2*P_0=0.390625*0.37583452=0.14681036`
`P_3=(0.625)^3*P_0=0.24414062*0.37583452=0.09175647`
`P_4=(0.625)^4*P_0=0.15258789*0.37583452=0.0573478`
`P_5=(0.625)^5*P_0=0.09536743*0.37583452=0.03584237`
`P_6=(0.625)^6*P_0=0.05960464*0.37583452=0.02240148`
`P_7=(0.625)^7*P_0=0.0372529*0.37583452=0.01400093`
`P_8=(0.625)^8*P_0=0.02328306*0.37583452=0.00875058`
`P_9=(0.625)^9*P_0=0.01455192*0.37583452=0.00546911`
`P_10=(0.625)^10*P_0=0.00909495*0.37583452=0.0034182`
This material is intended as a summary. Use your textbook for detail explanation.
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