Queuing Theory, M/M/s/N/N Queuing Model (M/M/c/K/K) calculator

SolutionHelp

1. Arrival Rate

$lambda=30$ , Service Rate

$mu=20$ , Number of servers

$s=2$ , Capacity

$N=3$

2. Arrival Rate

$lambda=10$ , Service Rate

$mu=3$ , Number of servers

$s=2$ , Capacity

$N=3$

3. Arrival Rate

$lambda=40$ , Service Rate

$mu=1$ , Number of servers

$s=10$ , Capacity

$N=10$

4. Arrival Rate

$lambda=45$ , Service Rate

$mu=15$ , Number of servers

$s=2$ , Capacity

$N=12$

5. Arrival Rate

$lambda=1/10$ , Service Rate

$mu=1/4$ , Number of servers

$s=2$ , Capacity

$N=5$

6. Arrival Rate

$lambda=1/10$ , Service Rate

$mu=1/4$ , Number of servers

$s=2$ , Capacity

$N=5$

Example

1. Queuing Model = mmsnn, Arrival Rate $lambda=30$ per 1 hr, Service Rate $mu=20$ per 1 hr, Number of servers $s=2$ , Limited Customer $N=3$

Solution:
Arrival Rate

$lambda=30$ per 1 hr and Service Rate

$mu=20$ per 1 hr (given)

Queuing Model : M/M/s/N/N

Arrival rate

$lambda=30,$ Service rate

$mu=20,$ Number of servers

$s=2,$ Machine

$N=3$ (given)

1. Traffic Intensity
$rho=lambda/mu$

$=(30)/(20)$

$=1.5$

2. Probability of no customers in the system
$P_0=[sum_{n=0}^(s-1) (N!)/((N-n)!*n!)*rho^n + sum_{n=s}^(N) (N!)/((N-n)!*s!*s^(n-s))*rho^n]^(-1)$

$=[sum_{n=0}^(1) (3!)/((3-n)!*n!)*(1.5)^n + sum_{n=2}^(3) (3!)/((3-n)!*2!*2^(n-2))*(1.5)^n]^(-1)$

$=[(1+(3!)/(2!*1!)*(1.5)^1) + ((3!)/(1!*2!*2^(0))*(1.5)^2+(3!)/(0!*2!*2^(1))*(1.5)^3)]^(-1)$

$=[(1+(3)/(1)*(1.5)^1) + ((3xx2)/(2*1)*(1.5)^2+(3xx2xx1)/(2*2)*(1.5)^3)]^(-1)$

$=[(1+4.5) + (6.75+5.0625)]^(-1)$

$=[17.3125]^(-1)$

$=0.05776173$ or

$0.05776173xx100=5.776173%$

3. Probability that there are n customers in the system
$P_n={((N!)/((N-n)!*n!)*rho^n*P_0, "for "0<=n< s),((N!)/((N-n)!*s!* s^(n-s))*rho^n*P_0, "for "s<=n<= N):}$

$P_n={((3!)/((3-n)!*n!)*(1.5)^n*P_0, "for "0<=n<2),((3!)/((3-n)!*2!*2^(n-2))*(1.5)^n*P_0, "for "2<=n<=3):}$

$P_1=0.2599278$

$P_2=0.3898917$

$P_3=0.29241877$

4. Average number of customers in the system
$L_s=sum_{n=0}^(N) nP_n$

$=sum_{n=0}^(3) n*P_n$

$=0*P_0+1*P_1+2*P_2+3*P_3$

$=0*0.05776173+1*0.2599278+2*0.3898917+3*0.29241877$

$=1.91696751$

5. Average number of customers in the queue
$L_q=sum_{n=s+1}^(N) (n-s)P_n$

$=sum_{n=3}^(3) (n-2)*P_n$

$=1*P_3$

$=1*0.29241877$

$=0.29241877$

6. Effective Arrival rate
$lambda_e=lambda(N-L_s)$

$=30*(3-1.91696751)$

$=32.49097473$

7. Average time spent in the system
$W_s=(L_s)/(lambda_e)=(L_s)/(lambda(N-L_s))$

$=(1.91696751)/(32.49097473)$

$=0.059$ hr or

$0.059xx60=3.54$ min

8. Average Time spent in the queue
$W_q=(L_q)/(lambda_e)=(L_q)/(lambda(N-L_s))$

$=(0.29241877)/(32.49097473)$

$=0.009$ hr or

$0.009xx60=0.54$ min

9. Utilization factor
$U=(L_s-L_q)/s$

$=(1.91696751-0.29241877)/(2)$

$=0.81227437$ or

$0.81227437xx100=81.227437%$