Queueing System Provide a mean to estimate important measures of Highway Performance Travel time Speed Affects Roadway Design Required left-turn bay length Queueing System Components CEE 8207 Summer 2013 L#6 Queue

Arrival Process/Pattern Average Arrival Rate Statistical distribution of interarrival times or headways Traffic usually Random arrival Poisson Distribution CEE 8207 Summer 2013 L#6 Queue

Service Process/Pattern & # of Servers Average Service Rate vs . Service rate is . Distribution of Service Time # of Servers # of channels to perform service i.e – Grocery Cashier CEE 8207 Summer 2013 L#6 Queue

Server Capacity & Queue Discipline How much can . i.e. Queue discipline Means by which next customer/entity is selected FIFO – First In, First Out i.e: LIFO – Last In, First Out Priority CEE 8207 Summer 2013 L#6 Queue

Queue Process (cont’) Kendal’s Notation To describe Queue Process in easy way i / j / n i: arrival process j: service process n: # of servers For arrival process and service process If Random → M If Deterministic → D In Traffic Arrival Rate = . Service Rate = . CEE 8207 Summer 2013 L#6 Queue

CEE 8207 Summer 2013 L#6 Queue

Example of Queue Process Signalized Intersection Arrival Process/Pattern Vehicle, Random Arrival Service Process/Pattern Green Signal Timing (i.e; 1 veh/2sec when green) Service Channel, # of Servers # of lanes for corresponding movement System Capacity . Queue Discipline CEE 8207 Summer 2013 L#6 Queue

Queue Diagram CEE 8207 Summer 2013 L#6 Queue

Queue Example Find An incident occurs on a freeway Freeway Capacity before the incident : 4,000 veh/hour Constant Flow : 2,900 veh/hour of morning commute Due to the incident, 8:00 – Freeway closed 8:12 – Partial open 2,000 veh/hour 8:31 – Freeway fully open Assume D/D/1 Queue, FIFO STEP I – Determine Arrival Pattern STEP II – Determine Service Pattern STEP III – Cumulative Concept STEP IV – Queue Diagram Can find all the answers Find Time of Queue Dissipation Longest Queue Length Total Delay Average Delay per Vehicle Longest Wait of any vehicle CEE 8207 Summer 2013 L#6 Queue

Step I – Arrival Pattern We need to determine which one is the . . . in this problem set Constant . 2,900 veh/hour = . CEE 8207 Summer 2013 L#6 Queue

Step II – Service Pattern . . : Service Rate . Due to the incident 8:00 – 8:12 8:13 – 8:30 8:31 after CEE 8207 Summer 2013 L#6 Queue

Step III – Cumulative Concept Need to express Step I and II in Equation Arrival Pattern Service Pattern Remember that there is – CEE 8207 Summer 2013 L#6 Queue

Step IV – Queue Diagram CEE 8207 Summer 2013 L#6 Queue

Step IV (2) Queue Dissipation Point 3777.5 78.16 CEE 8207 Summer 2013 L#6 Queue

Step IV (3) Max Delay and Queue Length Max diff btw arrival and service lines B Longest Queue C A Longest Delay Longest Queue = Longest Delay = CEE 8207 Summer 2013 L#6 Queue

Step IV (4) Total Delay, Average Delay C Area btw arrival and service lines B A Total Delay = A & C : Triangle B: (Trapezoid) – (Triangle) Triangle Area = 0.5×(length)×(height) Trapezoid Area = 0.5×(height) ×(Length1 + Length2) Average Delay = Total Delay / total # of Vehicles = CEE 8207 Summer 2013 L#6 Queue