1. A Queue system based on the bank service
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Native simulation to verify Jackson theorem
Presented by
Yuyao Li
Shan Gao

2. Outline 1.Background 2. Simulation scenario 3.Results and analysis
4.Conclusion

3. BACKGROUND
Many surveys indicate that the waiting time for service is one of the problems that a customer most concern about.
More results show that a bank with a higher customer satisfaction also has a higher profit.
The bank queuing system efficiency problem not only affects the customer’s satisfaction, but also greatly affects the economic benefit of the bank
Queuing theory adopts a statistical study on the arrival and the service time of the service object and then gets relevant service indicators, such as waiting time, average queue length and busy period length, etc.

4. Simulation Scenario
Server1 and Server 2: Services for saving, withdrawing, bank statement
with average service time about 4 minutes per person
Server3: Services for opening account, insurance and the like, with average
service time about 8 minutes per person.

5. Simulation Scenario Jackson’s theorem:
For a queuing system have multiple queues, each queue can independently be treated as an M/M/1 system.
Where ni is the number of customers in the i-th queuing system.
Therefore the number of customers in the whole system can be the product result of the number of customers in each queue.

6. Simulation Scenario
Flow chart for programming

7. Results and analysis Analytical results: θ 1=0.5 λ=0.15 ρ1=θ 1/u1=0.6
So the probability that the system with i (i=0,1,2,3····) customers π(i):
π(0)=π(0,0,0)=(1- ρ1) (1- ρ2) (1- ρ3)=0.1216
π(1)=π(1,0,0)+π(0,1,0)+ π(0,0,1)=…=
π(2)=π(2,0,0)+π(0,2,0)+π(0,0,2)…=
π(3)=π(3,0,0)+π(0,3,0)+π(0,0,3)…=

8. Results and analysis
The average probability that the system with n customers (n=0,1,2···)
Compared with the analytical results, the error is small enough

9. Conclusion Jackson’s theorem is correct, the error is less than 1%.
Jackson’s theorem makes more convenient to analysis the realistic situation.
In order to improve the service quality of a bank, more server needed