RELATED RESOUCES:
This session addresses a topic that applies to many situations both on and off roadway. Queue length determination and analysis identifies the characteristics of a waiting line of vehicles or people. It makes use of the Poisson distribution of arrivals along with a negative exponential distribution of service times. The procedure is useful in customer service analyses of drive-in banking and fast-food restaurants; it is also used in predicting toll-booth activity and parking garage utilization. Information such as the average length of waiting time, length of a queue, and length of a storage lane can be easily computed. The procedure is limited to single-channel or single-queue analysis, since it requires little data acquisition to produce meaningful results. Procedures exist for multi-channel analysis of two or more drive-in windows, toll booths, or similar service bays that require minimal data acquisition to render useful results.
The equipment list for data collection procedure is a stopwatch (cell phone) and note-taking materials.
Choose a location to study and a time to collect data
The student group should choose a fast-food facility or drive-in bank that has only one operating window. Fast-food restaurants that have two successive windows—one used to take order and the next to receive food—are acceptable. As an alternative, the student group may wish to choose a toll booth. If so, the toll booth should be a single booth with only one queue. It is important that the traffic entering and exiting the service line can be easily counted. This observation should take place during a peak period when the service rate (the number of vehicles serviced per hour) is at, or near, its highest. The service rate to be determined should be the maximum service rate the facility can render—when the arrival rate is at its highest and workers are the most motivated. For some facilities, this may be a significant factor in obtaining data needed to determine the service rate. For others, the service time may be unaffected by the arrival rate and length of queue. The student group should be aware of the possible effect on the data quality.
Collect data and record values
Two sets of data will be obtained. One should address the service rate of the facility and the other the variation in the length of the queue. Service time is the time it takes one vehicle from the queue to be serviced by the facility and depart. For a bank, the service time starts when the vehicle stops at the speaker or window and ends when the vehicle starts to move. Using the data collection form attached to this handout, 50 service times will be collected. Collection of the queue length variations over the course of 60 minutes is obtained by recording the queue length at the end of each minute for one hour. Consecutive readings are required over this period, since the variability in length of the queue is to be compared to theoretical computed values. The student group starts timing and at the end of each minute observes and records the length of the queue, excluding any vehicle being serviced. Simultaneously, the number of vehicles arriving and entering the queue is recorded at the end of each 5-minute arrival.
Check your work before leaving the field
Before leaving the field, the data collection form should be checked to see that all field data have been collected correctly. Any unusual observations should be listed as comments.
Summarize field data on form provided
Since the values were computed in the field, the computations are rechecked. Next, the queue-length data is reduced, using the data summary form. The procedure begins with identifying the number of queues that had 0 vehicle, 1 vehicle, 2 vehicles, and so on, and entering these values in Column 2. The sum of Column 2 should be 60, the number of queues observed. Column 3 is the frequency of that queue length and is computed by dividing Column 2 by the total, 60. Column 4, cumulative frequency, is computed by adding all frequencies, including the time interval.
Calculate standard relationships
Using the collected data, a number of standard relationships can be calculated. These can be used to identify and assess the performance characteristics of the facility. The formulae and an example calculation of each are shown below. These use an assumed service rate of 140 vehicles per hour and an assumed arrival rate of 90 vehicles per hour.
For each of the calculations, q equals the arrival rate in vehicles per hour, and Q equals the service rate in vehicles per hour.
The probability of exactly n units being in the system, where n equals the total number of vehicles—including the unit being served is:
\(Pr(n)=(\frac{q}{Q})^n(1-\frac{q}{Q})\)
Therefore, the probability of exactly 2 vehicles in the system is \(0.148\). The expected, or average number of vehicles in the system, \(E(n)\) is:
\(E(n)=\frac{q}{(Q-q)}\)
Therefore, the average number of vehicles in the system for the example is \(1.8\).
The average queue length, E(m),is:
\(E(m)= \frac{q^2}{Q(Q-q)}\)
The mean queue length is 1.16 vehicles.
The average waiting time in the queue after arrival, E(w), is:
\(E(w)=\frac{q}{Q(Q-q)}\)
This calculates to be \(0.013\) hours or \(46.3\) seconds.
The average time a vehicle spends in the system including being served and waiting to be served, E(v), is:
\(E(v)=\frac{1}{(Q-q)}\)
This calculates to be \(0.02\) hours or \(72\) seconds.
The probability of a vehicle spending time, t hours, or less in the system is:
\(Pr(v \le t)= 1- e^{-(1-\frac{q}{Q})qt}\)
If it is desirable to know how often the total time a vehicle will be in the system less than \(50\) seconds, then the equation yields \(0.36\).
The probability of a vehicle spending time, t, or less in the queue is:
\(Pr(w \le t) = 1-\frac{q}{Q}e^{-(1-\frac{q}{Q})qt}\)
Likewise, if it is desirable to know how often the waiting time will be less than or equal to \(50\) seconds, then the equation yields \(0.589\).
The probability of more than N vehicles in the queue is:
\(Pr(n \gt N)= (\frac{q}{Q})^{N+1}\)
This equation can be used to find the length of the storage lane for the queue. If it is desirable to have the storage lane long enough to accommodate 90% of the queues that develop, then Pr(n>N)=0.10 and the equation can be solved for N. For the example rates, it is computed that more than 4 vehicles will be in the queue 10.9% of the time, and more than 5 vehicles will be in the queue 7% of the time. Based on this, a storage lane long enough to accommodate 5 vehicles should be sufficient over 90% of the time.
Class demo
Summarize and interpret results
Having computed all of the relationships, the student group can now examine the results and draw conclusions about the expected operation of the facility. Among additional questions that may be addressed are:
Does the observed average queue length compare favorably to the computed value? If not, is there an indication why? If it is desirable for the facility to have sufficient storage length for 95% of the expected queues to be accommodated, how many vehicles must be accommodated? Do the computed values indicate the need for improvements in the operation of the facility, such as the addition of more windows or channels, or changes in the operation to increase the rate of service? Assuming an arrival rate equal to 90% of the service rate, how much of an increase in service time can occur at the facility before the queue length gets too long for the site?
These and other questions can provide useful input to the decision maker responsible for the operation of a business facility. Given the small amount of data collection needed to render significant results, the study of queue length is a cost-effective management tool. The procedure presented here for single-channel analysis is similar to that for multi-channel applications.