queuing theory (waiting lines)
DESCRIPTION
Forest Products Association (FPA) would like to find a way to reduce the waiting time for trucks delivering its wood to chipping facilities and has requested an evaluation and the new average waiting times for two suggested options, option #2 and option #3. As this report will show, option #2, that only includes the use of two chipping stations, is not effective since it will actually maximize wait time due to full utilization. Option #3, with the addition of a third chipper, is the better option since it will cut the average waiting time in line down to under 2 hours. There are also a couple of other ideas proposed. First, the possible application of the multiple‐server system to the FPA situation was explored. Another suggested improvement of option #3 is to make the wait times for each truck more fair by taking into account how much time each truck will need in order to be serviced.TRANSCRIPT
Forest Products Association Mini Case
Using Queuing Theory to Minimize Waiting Lines
Jesse Kedy
Forest Products Association (FPA) would like to find a way to reduce the waiting time for trucks delivering its wood to chipping facilities and has requested an evaluation and the new average waiting times for two suggested options, option #2 and option #3. As this report will show, option #2, that only includes the use of two chipping stations, is not effective since it will actually maximize wait time due to full utilization. Option #3, with the addition of a third chipper, is the better option since it will cut the average waiting time in line down to under 2 hours. There are also a couple of other ideas proposed. First, the possible application of the multiple‐server system to the FPA situation was explored. Another suggested improvement of option #3 is to make the wait times for each truck more fair by taking into account how much time each truck will need in order to be serviced.
© 2006, Jesse Kedy Operations Management www.jessekedy.net University of Richmond
Queuing theory and “waiting-in-line” calculations can yield information that affects many
variables in business operations. Any organization concerned with efficiency and with lowering costs
must take waiting times into account within operations. Think of any dining establishment with a resister
line, or any grocery store. However, before using waiting line models, it is necessary to prove or
disprove the main assumptions upon which they are based. Some assumptions of single-server waiting
line models are that the customer population is infinite, and that the length of the line is unlimited. The
population assumption applies here, as there are over 30 customers. According to this mini-case though,
the key assumptions that must be proven are that the arriving trucks follow a Poisson distribution pattern
and that the chipping service time follows a negative exponential distribution pattern.
In this case, the yard office provided data for arrivals and service times for a typical work day,
and it is assumed that this data closely matches that of an average work day. To determine whether or
not the Poisson distribution assumption is valid for Forest Products Association (FPA), the arrival times
of all thirty-five loads of wood during a twelve-hour work day have been visualized (Figure 1). Note the
fact that arrivals are not evenly spread.
This data has also been graphed (Figure 2), which looks like a typical Poisson distribution. To
create this graph, the twelve-hour work day was divided into twenty-four half-hour time intervals. Based
on this data, the number of trucks that arrived during each of the twenty-four time intervals was counted.
For example, during the first four time intervals, four, three, zero, and one trucks arrived in order. Zero,
one, two, three, or four trucks arrived in each of the intervals, and the relative frequency of each of those
interval arrival amounts was then calculated and displayed in Figure 2. As the graph shows, in some half
hour intervals there was only one arrival, in others two, three, or four, and in others none at all. With
Poisson distributions, arrivals are random and interval arrival rates vary amongst separate time intervals.
Thus, the truck arrival pattern in this case can be assumed to be a Poisson distribution.
As a rule, if arrival rates follow a Poisson distribution, the interval time between arriving trucks
is exponential. With a negative exponential distribution, the lengths of the actual service times vary;
some can be close to zero while others require a relatively long time. Looking at the data from the yard
office, the service times do vary. In addition, the Poisson distribution and negative exponential
distribution are alternate ways of presenting the same information. Therefore, having shown a Poisson
arrival distribution, there must be a negative exponential distribution of service times. With these two
assumptions, the waiting line models and equations can be accurately applied.
1
To discuss option #2 and option #3 of this case, single-server equations can be used since both
options use service models in which there is only one server per line.
Option # 2
Here, the calculated arrival rate (λ) for each line is 1.522 trucks per hour, and the calculated
service rate (μ) for each server is 1.5 trucks per hour.
Therefore, the utilization rate (ρ) is approximately equal to 1.0: there is approximately 100%
utilization in the system. This is a problem since as ρ approaches 1.0 (the utilization rate approaches
100%), average waiting time approaches infinity. In short, with option #2, waiting time is at a
maximum. Another way to explain the problem with option #2 is with the fact that the arrival rate is
greater than the service rate. Therefore, the server cannot handle the arrival rate, and the system will
eventually fail.
With the figures from option #2, some variables such as Ls (the number of customers in the
system) cannot even be calculated. For example, with the case of Ls, since μ is greater than λ, Ls would
be negative. Due to these problems, option #2 is not at all recommended.
Option # 3
Option #3, the addition of another chipping facility, is much better than option #2. Since this
option works better and since all its variable values can be found, they have been calculated and listed in
Figure 3. Here, the arrival rate is less than the service rate, and the system utilization is .676. That
utilization rate is a great improvement from the full utilization with option #2.
With option #3 and the three machines being busy about two thirds of the time, the average
waiting time in line (wq) is approximately 1.393 hours. This amount of waiting time is clearly preferable
to the near infinite wait time calculated for option #2.
However, a total cost analysis cannot be performed without more information on labor costs,
including the cost of operating the chipping facilities, the cost of workers, and the waiting cost incurred.
Other Possibilities
As an alternative, we can calculate the difference in waiting time with a single line, multiple
server (M=3) model. Clearly, this is not currently a possibility; still, the goal is to see how substantial
the difference in waiting time would be if it were possible.
As Figure 4 shows, the average waiting time in line would be reduced to .3145, or 18.87 minutes.
In theory, building the chipping facilities in a triangular shape could enable the third station to be at a
minimum distance from the first two (Figure 4.1). This would necessitate new safety and sound-
2
3
reduction measures, as depicted by the line barriers in the figure. If this was possible, a single line could
form as close to all three stations as possible, enabling the next driver in line to see all three stations and
be able to arrive quickly at the next open station. The idea here is to capitalize on the advantages of a
single-line model while avoiding the safety limitations. This may involve costly improvements to the
existing barriers.
It has already been made clear that option #3 is better than option #2, but one problem that still
exists with this option is that trucks with varying service times all must wait for approximately the same
amount of time. This could cause more complaints from truckers who have lighter loads (therefore
shorter service times) since they would have to wait as long as truckers who need much longer service.
According to the psychology of waiting, unfair waits seem longer; this could be a problem with
the current model. Another possibility to improve option #3 can be taken from the grocery store
industry. Here, we could create a fast lane aimed at trucks with lighter loads and shorter service times.
With this option, trucks with service times under 40 minutes would use the fast lane, while larger
loads would use the remaining 2 lanes. Figures 5 and 6 show calculated values for both the fast lane and
the remaining lanes.
Comparing the two charts, in the fast lane, the arrival rate is higher but the service rate is also
much higher, so the average waiting time line (wq) drops to 1.01 hours and an average of only 1.47
hours in the system (ws).
On the other hand, as shown in Figure 6, in the remaining 2 lanes, the arrival rate is lower but
the service rate is also much lower; here, the average time in line is 1.76 hours. Overall, there would be
a shorter average waiting time for those in the fast lane but a longer average waiting time for those in the
remaining lanes. Using 40 minutes as the cut-off service time for the fast lane, about fifty percent of the
trucks would end up in the fast lane. All this averages out to approximately the same total average
waiting time in line (wq) for the system. However, this could still be a better option than option #3 since,
as previously stated, fair waits seem shorter than unfair waits.
Figure 1: Truck Arrivals
1 21 41 61 81 101 121 141 161 181 201 221 241 261 281 301 321 341 361 381 401 421 441 461 481 501 521 541 561 581 601 621 641 661 681 701
Arrival time (minute)
Figure 2: Poisson Distribution (rate)
0.167
0.458
0.208
0.083 0.083
0.000
0.100
0.200
0.300
0.400
0.500
0 1 2 3 4Arrivals per 1/2 hour
Relative frequency
Figure 3: Single Channel Waiting Line Model Arrival rate λ = 1.0144928 Increment Δλ = 1 Interarrival Time 1/λ = 0.9857 Service rate μ = 1.5 Increment Δμ = 0.1 Service time 1/μ = 0.6667 Exponential Service Time System Utilization ρ = 0.6763 Probability system is empty P0 = 0.3237 Average number in line Lq = 1.4132 trucks Average number in system Ls = 2.0896 trucks Average time in line Wq = 1.3930 hrs Average time in system Ws = 2.0597 hrs
Trucks/half hr Occurrence Frequency0 4 0.167 1 11 0.458 2 5 0.208 3 2 0.083 4 2 0.083
Figure 4: Multiple Channel Waiting Line Model Arrival rate λ = 3.0434783
Service rate μ = 1.5 Increment Δλ = 0.1 Increment Δμ = 0.1
Interarrival Time 1/λ = 0.3286 Service time 1/μ = 0.6667 Number of servers (max 12) M = 3
System Utilization ρ = 0.6763 Probability system is empty P0 = 0.1065 Probability arrival must wait Pw = 0.4581 Average number in line Lq = 0.9573 Average number in system Ls = 2.9863 Average time in line Wq = 0.3145 Average time in system Ws = 0.9812 Average waiting time Wa = 0.6866
Figure 6: Single Channel Waiting Line Model (Other 2 Lanes)
Arrival rate λ = 0.7826087 Increment Δλ = 1 Interarrival Time 1/λ = 1.2778 Service rate μ = 1.1650485 Increment Δμ = 0.1 Service time 1/μ = 0.8583 Exponential Service Time System Utilization ρ = 0.6717 Probability system is empty P0 = 0.3283 Average number in line Lq = 1.3746 Average number in system Ls = 2.0464 Average time in line Wq = 1.7565 Average time in system Ws = 2.6148
Figure 5: Single Channel Waiting Line Model (“Fast Lane”)
Arrival rate λ = 1.4782609 Increment Δλ = 1 Interarrival Time 1/λ = 0.6765 Service rate μ = 2.1564482 Increment Δμ = 0.1 Service time 1/μ = 0.4637 Exponential Service Time System Utilization ρ = 0.6855 Probability system is empty P0 = 0.3145 Average number in line Lq = 1.4942 Average number in system Ls = 2.1797 Average time in line Wq = 1.0108 Average time in system Ws = 1.4745
Figure 4.1: Loaded trucks wait in line & approach chipping facility when idle (or nearly done with a preceding truck). Goal: to mimic the single-line, multiple-server model as closely as possible. Empty trucks then leave the area (see above).