Queuing Analysis Two analytical techniques can be employed to study queuing processes: Shock wave analysis Demand-capacity process is deterministic Suited to evaluating the space occupied by the queuing processes and their interactions Queuing analysis Deterministic (input-output) or stochastic Vertical queues are assumed

Objectives 1. To estimate queuing characteristics when demand exceeds capacity at a bottleneck. For example, a. Average number of vehicles in a queue at a given time b. Average delay per vehicle c. Total hours of delay d. Duration of queuing 2. To determine idle time (of system, employees, etc.)

Queuing Process Queuing system Arrivals Departures Queue Server Stream of customers who demand service Line of customers waiting to be served

Queuing Application Examples Bank teller problem Transportation applications Toll booth Check-in counter at airport Blockage on highway Signal at intersection

Queuing Analysis in Traffic Applications Arrivals Departures Queue Server arrival rate determined by traffic demand departure rate determined by facility capacity How long the queue is going to be? How much delay? Duration of congestion? …

Queuing Characteristics/Classifications Arrivals: l (rate, distribution) Servicing: m (rate, distribution) Queue discipline: (e.g.FIFO) Number of channels (m) Arrival rate=demand=input Service rate=capacity=departure=output Served in random order

Queue discipline First come (in) – first served (out) (FIFO) First in, last out (FILO) Random service (SIRO: served in random order) Priority Inpatient customer Infinite (∞)or finite lines

Queuing Models Deterministic Stochastic Arrival and servicing are uniform (same interval, no randomness) arrival and service rates may vary over time Can be used to temporary blockage, temporary overloading, and periodical interruption such as signal Stochastic constant long-term arrival and service rates Arrival and service conform to certain random distribution short-term random fluctuation causes queue (even with arrival rate <service rate)

Deterministic Queuing arrival l Slope: flow rate 1 Cumulative vehicles Point of queue accumulation m2 Point of queue dissipation Waiting time(delay) 1 Queue length departure m1 1 Time

Deterministic Queuing Example Vehicles arrive at the entrance to a park with a single gate where a ranger distributes a free brochure. Park opens at 8:00 a.m. and arrivals are 480 veh/hr. After 20 min, arrival rate declines to 120 veh/hr for remainder of day. Time required to distribute a brochure is 15 seconds. Describe operational characteristics.

Arrival and Departure Rates l(1) = 480 veh/hr / 60 min/hr = 8 veh/min for t < 20 l(2) = 120 veh/hr / 60 min/hr = 2 veh/min for t > 20 m= 60 sec/min / 15 sec/veh = 4 veh/min for all t

Total number of vehicles 10 30 20 40 50 60 300 250 200 150 100 Total number of vehicles Time

Total vehicle arrivals @ t = 20 10 30 20 40 50 60 300 250 200 150 100 Total number of vehicles Time (min) Total vehicle arrivals @ t = 20 8t = 8 (20) = 160 veh (20, 160) Arrival curve 1 8 1

Total vehicle arrivals @ t > 20 10 30 20 40 50 60 300 250 200 150 100 Total number of vehicles Time Total vehicle arrivals @ t > 20 160 + 2(t - 20) Arrival curve 2 2 1 (20, 160) Arrival curve 1

Vehicle departures @ all t 10 30 20 40 50 60 300 250 200 150 100 Total number of vehicles Time Arrival curve 2 (20, 160) 4 Arrival curve 1 1 Departure curve Vehicle departures @ all t 4t

Vehicle departures @ all t 10 30 20 40 50 60 300 250 200 150 100 Total number of vehicles Time Arrival curve 2 Point of queue dissipation (20, 160) Arrival curve 1 Departure curve Vehicle departures @ all t 4t

Total number of vehicles 10 30 20 40 50 60 300 250 200 150 100 Total number of vehicles Time Arrival curve 2 Point of queue dissipation (20, 160) Arrival curve 1 Departure curve arrivals = departures 160 + 2(t-20) = 4t t = 60 min

Total number of vehicles 10 30 20 40 50 60 300 250 200 150 100 Total number of vehicles Time (60, 240) Arrival curve 2 Point of queue dissipation (20, 160) Arrival curve 1 Departure curve Total vehicles 4t = 4(60) = 240 veh =8(20)+2(40)

Total number of vehicles 10 30 20 40 50 60 300 250 200 150 100 Total number of vehicles Time (60, 240) Arrival curve 2 Point of queue dissipation (20, 160) Longest vehicle delay (FIFO) Arrival curve 1 Departure curve Longest vehicle queue

What is Longest vehicle queue? 10 30 20 40 50 60 300 250 200 150 100 Total number of vehicles Time What is Longest vehicle queue? Arrival curve 2 Departure curve (20, 160) Longest Vehicle Queue Occurs at t = 20 Queue = Arrivals – Departures = 8t – 4t = 8(20) – 4(20) = 160 – 80 =80 vehs Arrival curve 1

What is Queue Length at t=40? 10 30 20 40 50 60 300 250 200 150 100 Total number of vehicles Time What is Queue Length at t=40? Arrival curve 2 Arrivals Departures Arrival curve 1 Veh. In Queue = Arrivals – Departures = [160 + 2(t-20)] – 4t =[160 + 2(40-20)] - 4(40) = 40 vehs Departure curve

Total number of vehicles 10 30 20 40 50 60 300 250 200 150 100 Total number of vehicles Time Arrival curve 2 Departure curve (20, 160) 160 = 8 tEnter tEnter = 20 min 160 = 4tExit tExit = 40 min Arrival curve 1 Longest vehicle delay Delay = tExit – tEnter Delay = 40 min – 20 min = 20 mins

Total number of vehicles 10 30 20 40 50 60 300 250 200 150 100 Total number of vehicles Time What is delay of the 200th vehicle? Delay = tExit – tEnter Delay = 50 min – 40 min = 10 mins Arrival curve 2 Departure curve Arrival curve 1 200 = 160 – 2(tEnter - 20) tEnter = 40 min 200 = 4tExit tExit = 50 min

Total number of vehicles 10 30 20 40 50 60 300 250 200 150 100 Total number of vehicles Time What is the TOTAL amount of Delay? 20 min 40 min 80 vehs

Total number of vehicles 10 30 20 40 50 60 300 250 200 150 100 Total number of vehicles Time What is the average delay and queue length? 240 Total delay=2400veh mins Little’s Formula

Deterministic Queuing Example: Signalized Intersection Uniform Delay Calculation Uniform delay: delay due to red time under uniform arrivals

Signalized Intersection

Queuing Patterns

More Examples (with Equations) LTs: Longest queue Ts: time when LTs occurs Tc: Total duration of congestion D: total delay W: average waiting time

Microscopic Analysis Deterministic queuing analysis can be undertaken at the microscopic level Requires arrival and departure times of individual vehicles Computer simulation often used to calculate MOEs

Microscopic Analysis Example: Signalized intersection (Left turn) Arrival to downstream is assumed to be 10secs after arrivals to upstream, which is the start of the left-turn lane and beyond the back of queue.

Stochastic Queuing Kendall’s notation Letter 1 / Letter 2 / Number Letter 1: arrivals process Letter 2: service process Number of channels (servers)

Stochastic Queuing Traffic Density ρ = λ/µ, where ρ < 1 (λ < µ) ρ = traffic intensity λ = mean arrival rate (vehicles per time interval) µ = mean service rate per channel (vehicles per time interval) λ and µ must have the same units

Kendall’s Notation Letter 1 / Letter 2 / Number M: Random (negative exponential distribution) D: Deterministic E: Erlang distribution G: Generalized arrival or departure Number: the number of channels Example: M/D/1 [or M/D/1(∞,FIFO)], M/M/1, M/M/n

M/D/1 Queuing Regime Arrival times are exponentially distributed (arrivals on Poisson distribution) Service rate deterministic (no random variation) One service channel

M/D/1 Queuing Traffic Intensity: Average number in system: (waiting and service) Average Waiting Time: Average time in system:

M/M/1 Queuing Arrivals and departures are exponentially distributed One service channel

M/M/1 Queuing Regime Traffic Intensity: Average Queue Length: Average Waiting Time: Average time in system :

M/M/1 Queuing Regime

Little’s Law Lq = lWq Or E(m)=lE(w)

Multichannel M/M/n

M/M/1 Example Park scenario. Arrivals: negative exponentially distributed, arrival rate = 180 vph (3 vehs/minute) Park ranger takes an average of 15 sec to distribute brochures. The service time distribution is also negative exponential.

M/M/1 Example

Stochastic Queuing Example: Random Delay at Intersection Classical Queuing System Applications: Poison Arrival, General (Arbitrary, but continuous) service times (M,G,1) Poisson Arrival, Erlang service times (M, Ea,1) Poison Arrival, Uniform service times (M,D,1)