The city of Beljing is studying it Toursits Information Center. The Information collected about tourists arriving at the center is:
interval time ( time between Arrivals)in minutes:(3,4,5,6,7)Probabilty percentage(20.2,23.6,31.2,18.4,6.6)
The tourist receive service at one desk the follwoing distrubiton:Length of service (minutes)(2,3,4,5)Probability (percent)(15.6,28.7,36.2,19.5)
The center is opened for business at 9a, the firsttourist arrives at 9 am.
Simulate for 12 tourists to estimate(answer all questions by simulation)
The average wating time prior to service.
The average numberof tourist in the information center
The percentage of idle time at the center.
The probabilty of finding more that one tourist at the center.
Use the following RN( start assigning ranges with 000)
For service:
(826,958,489,643,781,321,590,187,002,608,412,120)
For arrivals:(242,318,876,408,630,027,716,903,530,297,802,596)
To begin the simulation, we need to generate random numbers to determine the arrival times of the tourists and the service times.
1. Generate Arrival Times:
- Assign ranges to the given probabilities for interval time and generate random numbers using the provided RN for arrivals.
- For example, for the interval time (3 minutes) with a probability of 20.2%, the range would be 000-020.2. If a random number falls within this range, it means the next tourist arrives after 3 minutes.
- Repeat this process for all interval times and generate random arrival times for 12 tourists.
2. Generate Service Times:
- Assign ranges to the given probabilities for length of service and generate random numbers using the provided RN for service.
- For example, for the length of service (2 minutes) with a probability of 15.6%, the range would be 000-015.6. If a random number falls within this range, it means the service time will be 2 minutes.
- Repeat this process for all service times and generate random service times for each tourist.
3. Calculate Waiting Time:
- Calculate the waiting time for each tourist by subtracting the arrival time from the service time.
4. Calculate Average Waiting Time:
- Sum up all the waiting times and divide by the total number of tourists (12) to find the average waiting time.
5. Calculate Average Number of Tourists in the Information Center:
- Count the number of tourists present in the information center at each arrival time and divide by the total number of arrivals to find the average number of tourists in the information center.
6. Calculate Percentage of Idle Time:
- Calculate the total idle time by subtracting the sum of all service times from the total simulation time (12 tourists). Divide this idle time by the total simulation time and multiply by 100 to find the percentage of idle time.
7. Calculate Probability of Finding More Than One Tourist at the Center:
- Count the number of arrivals with more than one tourist present in the information center and divide by the total number of arrivals to find the probability.
Perform these steps iteratively for the 12 simulated tourists and calculate the required values.
To simulate the system for 12 tourists and estimate the required values, we will use the provided random numbers to generate the inter-arrival times and service times.
1. Generate the Arrival Times:
- Start with the first random number 242.
- Find the corresponding interval based on the given probabilities. In this case, it falls in the interval (3 minutes).
- Record the first arrival time as 9 am.
- Add the interval time (3 minutes) to the first arrival time to get the second arrival time.
- Repeat this process for the remaining random numbers to generate all 12 arrival times.
2. Generate the Service Times:
- Use the provided random numbers to generate the service times for each tourist.
- Assign the intervals (2, 3, 4, 5) to the corresponding random numbers based on the given probabilities.
- Repeat this process for all 12 tourists to get their service times.
3. Calculate the Average Waiting Time Prior to Service:
- Subtract the arrival time from the previous arrival time to get the waiting time for each tourist.
- Sum up all the waiting times and divide by the total number of tourists (12) to get the average waiting time.
4. Calculate the Average Number of Tourists in the Information Center:
- Start with the number of tourists in the center at 9 am (which is 0).
- For each arrival time, if there is no ongoing service, increment the number of tourists in the center by 1.
- If there is an ongoing service at the time of arrival, increment the number of tourists by 1 but also decrement it by 1 at the completion of the service time.
- At each time point, record the number of tourists in the center.
- Sum up all the recorded numbers and divide by the total number of time points to get the average number of tourists.
5. Calculate the Percentage of Idle Time at the Center:
- Calculate the total time the center is idle by subtracting the sum of all service times from the total simulation time (12 arrivals * 5 minutes).
- Divide the total idle time by the total simulation time and multiply by 100 to get the percentage of idle time.
6. Calculate the Probability of Finding More Than One Tourist at the Center:
- Count the number of time points where the number of tourists in the center is more than one.
- Divide the count by the total number of time points and multiply by 100 to get the probability.
By following these steps and using the provided random numbers and probabilities, you can simulate the system and estimate the required values.