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In this model the arrival and service rates are deterministic that is the
arrival and service times of each vehicle are known.
- Customers are assumed to be patient.
- System is assumed to have unlimited capacity.
- Users arrive from an unlimited source.
- The queue discipline is assumed to be first in first out.
Morning peak traffic upstream of a toll booth is given in the table 2.
The toll plaza consists of three booths, each of which can handle an average of
one vehicle every 8 seconds.
Determine the maximum queue, the longest delay to an individual vehicle.
| Time period |
10 min volume |
| 7.00-7.10 |
200 |
| 7.10-7.20 |
400 |
| 7.20-7.30 |
500 |
| 7.30-7.40 |
250 |
| 7.40-7.50 |
200 |
| 7.50-8.00 |
150 |
The arrival volume is given in the table.
Service rate is given as 8 seconds per vehicle.
This implies for 10 min, 75 vehicles can be served by each server.
It is given there are 3 servers.
Hence 225 vehicles can be served by 3 servers in 10 min.
In the first 10 min only 200 vehicles arrive which are served so the service
rate for rest 50 min is 225 veh/10 min as there is a queue for the rest period.
The solution to the problem is showed in the table 3 following.
The cumulative arrivals and services are calculated in columns 3 and 5.
Queue length at the end of any 10 min interval is got by simply subtracting
column 5 from column 3 and is recorded in column 6.
Maximum of the column 6 is maximum queue length for the study period which is
300 vehicles.
The service rate has been found out as 225 vehicles per hour.
From proportioning we get the time required for each queue length to be served
and as 475 vehicles is the max queue length, the max delay is corresponding to
this queue.
Therefore max delay is 21.11 min.
| Time |
10 min |
Cum. |
Service |
Cumulative |
Queue |
Delay |
| period |
flow (3) |
rate(4) |
service(5) |
=(3)-(4) |
(6) |
|
| 7.00-7.10 |
200 |
200 |
200 |
200 |
0 |
0 |
| 7.10-7.20 |
400 |
600 |
225 |
425 |
175 |
7.78 |
| 7.20-7.30 |
500 |
1100 |
225 |
650 |
450 |
20.00 |
| 7.30-7.40 |
250 |
1350 |
225 |
875 |
475 |
21.11 |
| 7.40-7.50 |
200 |
1550 |
225 |
1100 |
450 |
20.00 |
| 7.50-8.00 |
150 |
1700 |
225 |
1325 |
375 |
16.67 |
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