Queuing Theory, M/M/infinity Queuing Model calculator

SolutionHelp

1. Arrival Rate

$lambda=8$ , Service Rate

$mu=9$

2. Arrival Rate

$lambda=6$ , Service Rate

$mu=7$

3. Arrival Rate

$lambda=10$ per 8 hr, Service Rate

$mu=1$ per 30 min

4. Arrival Rate

$lambda=4$ per 1 hr, Service Rate

$mu=1$ per 10 min

5. Arrival Rate

$lambda=30$ per 1 day, Service Rate

$mu=1$ per 36 min

6. Arrival Rate

$lambda=96$ per 1 day, Service Rate

$mu=1$ per 10 min

Example

1. Queuing Model = mminf, Arrival Rate $lambda=8$ per 1 hr, Service Rate $mu=9$ per 1 hr

Solution:
Arrival Rate

$lambda=8$ per 1 hr and Service Rate

$mu=9$ per 1 hr (given)

Queuing Model : M/M/ $oo$

Arrival Rate

$lambda=8,$ Service Rate

$mu=9$ (given)

1. Traffic Intensity
$rho=lambda/mu$

$=(8)/(9)$

$=0.88888889$

2. Probability of no customers in the system
$P_0=e^(-rho)$

$=e^(-0.88888889)$

$=0.41111229$ or

$0.41111229xx100=41.111229%$

3. Probability that there are n customers in the system
$P_n=rho^n/(n!)*P_0$

$P_n=(0.88888889)^n/(n!)*P_0$

$P_1=((0.88888889)^1)/(1!)*P_0=0.88888889/1*0.41111229=0.36543315$

$P_2=((0.88888889)^2)/(2!)*P_0=0.79012346/2*0.41111229=0.16241473$

$P_3=((0.88888889)^3)/(3!)*P_0=0.70233196/6*0.41111229=0.04812288$

$P_4=((0.88888889)^4)/(4!)*P_0=0.62429508/24*0.41111229=0.01069397$

$P_5=((0.88888889)^5)/(5!)*P_0=0.55492896/120*0.41111229=0.00190115$

$P_6=((0.88888889)^6)/(6!)*P_0=0.49327018/720*0.41111229=0.00028165$

$P_7=((0.88888889)^7)/(7!)*P_0=0.43846239/5040*0.41111229=0.00003577$

4. Average number of customers in the system
$L_s=rho$

$=0.88888889$

5. Average number of customers in the queue
$L_q=0$

6. Average time spent in the system
$W_s=1/mu$

$=1/(9)$

$=0.11111111$ hr or

$0.11111111xx60=6.66666667$ min

7. Average Time spent in the queue
$W_q=0$