Queuing

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					            Queuing
Fall 2008
CEE 320




CEE 320
Anne Goodchild
            Outline

                      1.   Fundamentals
                      2.   Poisson Distribution
                      3.   Notation
                      4.   Applications
                      5.   Analysis
                           a. Graphical
                           b. Numerical
                      6. Example
Fall 2008
CEE 320
            Fundamentals of Queuing Theory

            • Microscopic traffic flow
              – Different analysis than theory of traffic flow
              – Intervals between vehicles is important
              – Rate of arrivals is important


            • Arrivals
            • Departures
            • Service rate
Fall 2008
CEE 320
            Activated
                     Upstream of bottleneck/server              Downstream


              Arrivals                                          Departures




                                            Server/bottleneck
Fall 2008
CEE 320




               Direction of flow
            Not Activated


               Arrivals          Departures




                            server
Fall 2008
CEE 320
            Flow Analysis

            • Bottleneck active
              – Service rate is capacity
              – Downstream flow is determined by bottleneck
                service rate
              – Arrival rate > departure rate
              – Queue present
Fall 2008
CEE 320
            Flow Analysis

            • Bottle neck not active
              –   Arrival rate < departure rate
              –   No queue present
              –   Service rate = arrival rate
              –   Downstream flow equals upstream flow
Fall 2008
CEE 320
            • http://trafficlab.ce.gatech.edu/freewayapp/
              RoadApplet.html
Fall 2008
CEE 320
            Fundamentals of Queuing Theory

            • Arrivals
              – Arrival rate (veh/sec)
                 • Uniform
                 • Poisson
              – Time between arrivals (sec)
                 • Constant
                 • Negative exponential
            • Service
              – Service rate
              – Service times
                 • Constant
Fall 2008
CEE 320




                 • Negative exponential
            Queue Discipline

            • First In First Out (FIFO)
              – prevalent in traffic engineering
            • Last In First Out (LIFO)
Fall 2008
CEE 320
            Queue Analysis – Graphical
                                           D/D/1 Queue
                                                                            Departure
                                                                              Rate
                            Delay of nth arriving vehicle                      Arrival
                                                                                Rate
                        Maximum queue
             Vehicles




                                                            Maximum delay

                                                   Total vehicle delay


                                 Queue at time, t1


                           t1                    Time
Fall 2008
CEE 320




                                                                     Where is capacity?
            Poisson Distribution
            • Good for modeling random events
            • Count distribution
              – Uses discrete values
              – Different than a continuous distribution


                                P n  
                                         t      n
                                                    e    t


                                                   n!
                 P(n) = probability of exactly n vehicles arriving over time t
                    n = number of vehicles arriving over time t
                    λ = average arrival rate
                    t = duration of time over which vehicles are counted
Fall 2008
CEE 320
            Poisson Ideas

            • Probability of exactly 4 vehicles arriving
               – P(n=4)
            • Probability of less than 4 vehicles arriving
               – P(n<4) = P(0) + P(1) + P(2) + P(3)
            • Probability of 4 or more vehicles arriving
               – P(n≥4) = 1 – P(n<4) = 1 - P(0) + P(1) + P(2) + P(3)

            • Amount of time between arrival of successive vehicles



              P0  Ph  t  
                                 t         0
                                              e    t
                                                          e t  e  qt 3600
                                             0!
Fall 2008
CEE 320
                                Example Graph
                                       0.25




                                       0.20
            Probability of Occurance




                                       0.15




                                       0.10




                                       0.05




                                       0.00
Fall 2008




                                              0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
CEE 320




                                                                          Arrivals in 15 minutes
                             Example Graph
                                       0.25


                                                                                                Mean = 0.2 vehicles/minute
                                       0.20
            Probability of Occurance




                                                                                                Mean = 0.5 vehicles/minute


                                       0.15




                                       0.10




                                       0.05




                                       0.00
Fall 2008




                                              0   1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20
CEE 320




                                                                          Arrivals in 15 minutes
                 Example: Arrival Intervals
                                       1.0

                                       0.9                                    Mean = 0.2 vehicles/minute
                                       0.8                                    Mean = 0.5 vehicles/minute
            Probability of Excedance




                                       0.7

                                       0.6

                                       0.5

                                       0.4

                                       0.3

                                       0.2

                                       0.1

                                       0.0
                                             0   2   4    6      8     10     12     14     16      18     20
Fall 2008
CEE 320




                                                         Time Between Arrivals (minutes)
            Queue Notation

                                                 Number of
             Arrival rate nature                 service channels


                                   X /Y / N
                                              Departure rate nature

            • Popular notations:
              – D/D/1, M/D/1, M/M/1, M/M/N
              – D = deterministic
              – M = some distribution
Fall 2008
CEE 320
            Queuing Theory Applications
            • D/D/1
              – Deterministic arrival rate and service times
              – Not typically observed in real applications but
                reasonable for approximations
            • M/D/1
              – General arrival rate, but service times
                deterministic
              – Relevant for many applications
            • M/M/1 or M/M/N
              – General case for 1 or many servers
Fall 2008
CEE 320
            Queue times depend on variability
Fall 2008
CEE 320
                                                         Steady state assumption

            Queue Analysis – Numerical
                                                                          
                                                                              1.0
                                                                          

            • M/D/1
                                                            2
              – Average length of queue             Q
                                                         21   

                                                   1   
              – Average time waiting in queue w     1  
                                                          
                                                  2      

                                                               1  2  
              – Average time spent in system               t     1  
                                                                       
                                                              2       

               λ = arrival rate μ = departure rate =traffic intensity
Fall 2008
CEE 320
            Queue Analysis – Numerical
                                                                         
                                                                             1.0
                                                                         

            • M/M/1
                                                         2
              – Average length of queue             Q
                                                       1   

                                                 1  
                                                     
              – Average time waiting in queue w        
                                                       

                                                                  1
              – Average time spent in system               t
                                                                 

               λ = arrival rate μ = departure rate =traffic intensity
Fall 2008
CEE 320
            Queue Analysis – Numerical
                                                                         
                                                                            N  1.0
                                                                         

            • M/M/N
                                                       P0  N 1    1     
              – Average length of queue             Q                    
                                                        N! N  1   N 2 

                                                   Q 1
              – Average time waiting in queue w      
                                                       

                                                                 Q
              – Average time spent in system               t
                                                                  

               λ = arrival rate μ = departure rate =traffic intensity
Fall 2008
CEE 320
            M/M/N – More Stuff
                                                                              
                                                                                 N  1.0
              – Probability of having no vehicles                             
                                       1
                         P0  N 1 nc
                                          N
                              0 n !  N!1   N 
                              nc    c


              – Probability of having n vehicles
                         P0
                         n                                n P0
                 Pn           for n  N          Pn        n N
                                                                         for n  N
                         n!                              N          N!

              – Probability of being in a queue
                                              P0  N 1
                                Pn N   
                                          N! N 1   N 
Fall 2008
CEE 320




               λ = arrival rate μ = departure rate =traffic intensity
            Poisson Distribution Example
            Vehicle arrivals at the Olympic National Park main gate are assumed
            Poisson distributed with an average arrival rate of 1 vehicle every 5
            minutes. What is the probability of the following:

                 1. Exactly 2 vehicles arrive in a 15 minute interval?
                 2. Less than 2 vehicles arrive in a 15 minute interval?
                 3. More than 2 vehicles arrive in a 15 minute interval?



                  Pn  
                          0.20 veh min  t            n
                                                            e  0.20 veh m int
                                                   n!
Fall 2008
CEE 320




                                                                                   From HCM 2000
            Example Calculations

              Exactly 2:   P2  
                                   0.20  15 2 e 0.20 15    0.224  22 .4%
                                               2!
            Less than 2:   Pn  2  P0  P1  0.1992

                           P(0)=e-.2*15=0.0498, P(1)=0.1494


            More than 2:   Pn  2  1  P0  P1  P2  0.5768
Fall 2008
CEE 320
            Example 1
            You are entering Bank of America Arena at Hec Edmunson Pavilion to
            watch a basketball game. There is only one ticket line to purchase
            tickets. Each ticket purchase takes an average of 18 seconds. The
            average arrival rate is 3 persons/minute.
            Find the average length of queue and average waiting time in queue
            assuming M/M/1 queuing.
Fall 2008
CEE 320
            Example 1

            • Departure rate: μ = 18 seconds/person or 3.33
              persons/minute
            • Arrival rate: λ = 3 persons/minute
            • ρ = 3/3.33 = 0.90

            • Q-bar = 0.902/(1-0.90) = 8.1 people
            • W-bar = 3/3.33(3.33-3) = 2.73 minutes

            • T-bar = 1/(3.33 – 3) = 3.03 minutes
Fall 2008
CEE 320
            Example 2
            You are now in line to get into the Arena. There are 3 operating
            turnstiles with one ticket-taker each. On average it takes 3 seconds
            for a ticket-taker to process your ticket and allow entry. The average
            arrival rate is 40 persons/minute.
            Find the average length of queue, average waiting time in queue
            assuming M/M/N queuing.
Fall 2008
CEE 320
            Example 2

            •   N=3
            •   Departure rate: μ = 3 seconds/person or 20 persons/minute
            •   Arrival rate: λ = 40 persons/minute
            •   ρ = 40/20 = 2.0
            •   ρ/N = 2.0/3 = 0.667 < 1 so we can use the other equations

            •   P0 = 1/(20/0! + 21/1! + 22/2! + 23/3!(1-2/3)) = 0.1111
            •   Q-bar = (0.1111)(24)/(3!*3)*(1/(1 – 2/3)2) = 0.88 people
            •   T-bar = (2 + 0.88)/40 = 0.072 minutes = 4.32 seconds
            •   W-bar = 0.072 – 1/20 = 0.022 minutes = 1.32 seconds
Fall 2008
CEE 320
            Example 3
            You are now inside the Arena. They are passing out Harry the Husky
            doggy bags as a free giveaway. There is only one person passing
            these out and a line has formed behind her. It takes her exactly 6
            seconds to hand out a doggy bag and the arrival rate averages
            9 people/minute.
            Find the average length of queue, average waiting time in queue, and
            average time spent in the system assuming M/D/1 queuing.
Fall 2008
CEE 320
            Example 3

            • N=1
            • Departure rate: μ = 6 seconds/person or 10
              persons/minute
            • Arrival rate: λ = 9 persons/minute
            • ρ = 9/10 = 0.9

            • Q-bar = (0.9)2/(2(1 – 0.9)) = 4.05 people
            • W-bar = 0.9/(2(10)(1 – 0.9)) = 0.45 minutes = 27
              seconds
            • T-bar = (2 – 0.9)/((2(10)(1 – 0.9) = 0.55 minutes =
              33 seconds
Fall 2008
CEE 320
            Primary References

            •   Mannering, F.L.; Kilareski, W.P. and Washburn, S.S. (2003). Principles
                of Highway Engineering and Traffic Analysis, Third Edition (Draft).
                Chapter 5

            •   Transportation Research Board. (2000). Highway Capacity Manual
                2000. National Research Council, Washington, D.C.
Fall 2008
CEE 320

				
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