# qlec6 - Queueing Systems Lecture 6 Amedeo R Odoni November...

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November 1, 2004 Queueing Systems: Lecture 6 Amedeo R. Odoni Lecture Outline Complete discussion of dynamic queues (qualitative obsrvations) Congestion pricing in transportation: the fundamental ideas Congestion pricing and queueing theory Numerical example A real example from LaGuardia airport Practical complications Reference: Handout on “Congestion Pricing and Queueing Theory” (on course website)

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Comparison of August Weekday Peaking Patterns 1993 vs. 1998 (3 Hour Average) Operations 130 120 110 1993 1998 100 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 1 01 11 21 31 41 51 61 71 81 92 02 12 22 Hour Two common “approximations” (??) for dynamic demand profiles 1. Find the average demand per unit of time for the time interval of interest and then use steady-state expressions to compute estimates of the queuing statistics. [Problems?] 2. Subdivide the time interval of interest into periods during which demand stays roughly constant; apply steady-state expressions to each period separately. [Problems?]
Problems with the Approximate Methods Problems with Approach 1: 1. For cases in which demand varies significantly (e.g., >10% from overall average value) the delay estimates can be VERY poor 2. Will underestimate overall average delay, possibly by a lot Problems with Approach 2: 1. May not have ρ < 1, for some intervals; then what? 2. Time to reach “steady state” is large for values of which are close to 1; therefore “steady state” expressions may be very poor approximations when intervals are relatively short 3. Approach does not take into consideration the “dynamics” of the demand profile The Two Viable Approaches 1. Simulation: High level of detail May be only viable alternative for complex systems Statistical significance of results? 2. Numerical solution of equations describing the evolution of queueing system over time: Increasingly practical May provide lots of information, such as P n (t)

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Dynamic Behavior of Queues and difficult to predict 1. The dynamic behavior of a queue can be complex 2. Expected delay changes non-linearly with changes in the demand rate or the capacity 3. The closer the demand rate is to capacity, the more sensitive expected delay becomes to changes in the demand rate or the capacity 4. The time when peaks in expected delay occur may lag behind the time when demand peaks 5. The expected delay at any given time depends on the “history” of the queue prior to that time 6. The variance (variability) of delay also increases when the demand rate is close to capacity 0 5 10 15 20 25 30 35 40 1:00 3 :00 5 7:00 9 :0 0 11 13: 0 15:0 1 7:0 19 21:00 23:0 R1 R2 R3 R4 i 30 15 45 60 75 90 ) The dynamic behavior of a queue; expected delay for four different levels of capacity Dem Delays (m ns) Demand (movements) 105 120 (R1= capacity is 80 movements per hour; R2 = 90; R3 = 100; R4 = 110
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qlec6 - Queueing Systems Lecture 6 Amedeo R Odoni November...

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