Question

Question 1

A camera is located 3km away from the entrance gate of a port to count the number of incoming trucks. Trucks that have passed the camera but have not passed the gate are considered to be in the queueing system, in which some trucks are waiting to be serviced by the gate and the other trucks are being serviced by the gate. If truck A passes the camera before truck B, it will pass the gate no later than truck B (first-come-first serve). If truck A passes the camera later than truck B, it will pass the gate no earlier than truck B. Suppose the port does not work at mid-night, and the gate opens at 7:00am. The cumulative numbers of trucks that have passed the camera and the gate at different times of a day are shown in the table below. For brevity, assume trucks pass the camera and the gate only at 6:00, 6:10, 6:20, etc. That is, suppose that in the table below 5 trucks passed the camera exactly at 6:10, and 10 trucks pass the gate exactly at 7:10.

Table Q1-1. Cumulative numbers of trucks that have passed the camera and the gate

Time	6:00	6:10	6:20	6:30	6:40	6:50
#camera	0	5	10	25	44	50
#gate	0	0	0	0	0	0
Time	7:00	7:10	7:20	7:30	7:40	7:50
#camera	60	62	81	88	95	105
#gate	0	10	20	35	55	75
Time	8:00	8:10	8:20	8:30	8:40	8:50
#camera	110	112	115	130	132	139
#gate	95	110	111	115	125	139

Q1.1: Calculate the average flow rate per hour of the system for the three hours from 6:00am to 8:59:59am.

Q1.2: (You can use software tools, e.g., Microsoft Excel, to help answer this question.) Calculate the average flow time of all flow units.

Q1.3: (You can use software tools, e.g., Microsoft Excel, to help answer this question.) Calculate the average inventory of the system from 6:00am to 8:59:59am, that is, the average number of trucks in the system at 6:00:01, 6:10:01, 6:20:01, 6:30:01, 6:40:01, … 8:40:01, 8:50:01.

Answer 1

Given: A system has a gate (service counter and depart immediately) and a camera (where a truck enters the system)
The service is one first come, first serve basis.
It is given that the distance between the camera and queue is 3Km.

Answer:
As the distance between the gate and camera is 3km, the maximum queue length is 3km.

The rate of flow is defined as the difference between rate of arrival and rate of departure.

The number of trucks arriving per hour are:
from 6 am to 7 am : 50 trucks (cumulative sum is given)
from 7 am to 8 am : 105-50=55 trucks
from 8 am to 9 am: 139-150=34 trucks

The number of trucks departing per hour are:
from 6 am to 7 am : 0  (cumulative sum is given)
from 7 am to 8 am : 75-0=75 trucks
from 8 am to 9 am : 139-75= 64 trucks

Ans 1.1) The rate of flow per hour is
from 6 am to 7 am : rate of arrival - rate of departure = 50-0 = 50 trucks
from 7 am to 8 am : rate of arrival - rate of departure = 55-75 = -20 trucks
from 8 am to 9 am : rate of arrival - rate of departure = 34-64 = -30 trucks

Thus the average rate of flow for three hours from 6 am to 9 am is (50-20-30)/3 which is 0.
we infer that all the trucks are given the service and they leave the system by 9 am.

Ans 1.2) The flow times for all flow units is given the image:

( The calculations are done using excel)

The average rate of flow is zero implying that all the trucks are given the service and they leave the system by 9 am.

Ans 1.3) The average inventory of the system from 6:00am to 8:59:59am, that is, the average number of trucks in the system at 6:00:01, 6:10:01, 6:20:01, 6:30:01, 6:40:01, … 8:40:01, 8:50:01. is given by: