Queuing Theory (waiting Lines)

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 www.jessekedy.net

Operations Management 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.

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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-

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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.

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Figure 1: Truck Arrivals

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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 3:

Figure 2: Poisson Distribution (rate) 0.458

0.500 0.400 Relative 0.300 frequency 0.200

0.208

0.167

0.083

0.100

Single Channel Waiting Line Model

Arrival rate Increment Interarrival Time

λ= Δλ = 1/λ =

1.0144928 1 0.9857

Service rate Increment Service time

μ= Δμ = 1/μ =

1.5 0.1 0.6667

0.083

ρ=

Exponential Service Time 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

0.000 0

Trucks/half hr 0 1 2 3 4

1

2 3 Arrivals per 1/2 hour

Occurrence Frequency 4 0.167 11 0.458 5 0.208 2 0.083 2 0.083

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System Utilization

Figure 4: Multiple Channel Waiting Line Model Arrival rate Increment Interarrival Time

λ= Δλ = 1/λ =

Number of servers (max 12) System Utilization Probability system is empty Probability arrival must wait Average number in line Average number in system Average time in line Average time in system Average waiting time

Service rate μ= Increment Δμ = Service time 1/μ =

3.0434783 0.1 0.3286 M= ρ= P0 = Pw = Lq = Ls = Wq = Ws = Wa =

1.5 0.1 0.6667

3 0.6763 0.1065 0.4581 0.9573 2.9863 0.3145 0.9812 0.6866

Figure 5: Single Channel Waiting Line Model (“Fast Lane”) Arrival rate Increment Interarrival Time Service rate Increment Service time

λ= Δλ = 1/λ = μ= Δμ = 1/μ =

1.4782609 1 0.6765 2.1564482 0.1 0.4637

ρ=

Exponential Service Time 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

System Utilization

Figure 6: Single Channel Waiting Line Model (Other 2 Lanes) Arrival rate Increment Interarrival Time Service rate Increment Service time

0.7826087 1 1.2778 1.1650485 0.1 0.8583

ρ=

Exponential Service Time 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

System Utilization

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).

λ= Δλ = 1/λ = μ= Δμ = 1/μ =