queueing_theory

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Unformatted text preview: Introduction to Transportation Engineering Introduction to Queueing Theory Dr. A.A. Trani Professor of Civil and Environmental Engineering Spring 2011 Virginia Polytechnic Institute and State University 1 of 95 Material Presented in this Section Topics Queueing Models + Background + Analytic solutions for various disciplines + Applications to infrastructure planning The importance of queueing models in infrastructure planning and design cannot be overstated. Queueing models offer a simplified way to analyze critical areas inside an airport terminal to evaluate levels of service and operational performance. Virginia Polytechnic Institute and State University 2 of 95 Principles of Queueing Theory Historically starts with the work of A.K. Erlang while estimating queues for telephone systems Applications are very numerous: •Transportation planning (vehicle delays in networks) •Public health facility design (emergency rooms) •Commerce and industry (waiting line analysis) •Communications infrastructure (switches and lines) Virginia Polytechnic Institute and State University 3 of 95 Elements of a Queue a) Input Source b) Queue c) Service facility Arriving Entities Input Source Queue Service facility Served Entities Queueing System Virginia Polytechnic Institute and State University 4 of 95 Specification of a Queue •Size of input source •Input function •Queue discipline •Service discipline •Service facility configuration •Output function (distribution of service times) Sample queue disciplines •FIFO - first in, first out •Time-based disciplines •Priority discipline Virginia Polytechnic Institute and State University 5 of 95 What Does a Queue Represent? Queues represent the state of a system such as the number of people inside an airport terminal, the number of trucks waiting to be loaded at a construction site, the number of ships waiting to be unloaded in a dock, the number of aircraft holding in an imaginary racetrack flight pattern near an airport facility, etc. The important feature seems to be that the analysis is common to many realistic situations where a flows of traffic (including pedestrians movind inside airport terminals) can be described in terms of either continuous flows or discrete events. Virginia Polytechnic Institute and State University 6 of 95 Types of Queues Deterministic queues - Values of arrival funtion are not random variables (continuous flow) but do vary over time. • Example of this process is the hydrodynamic approximation of pedestrian flows inside airport terminals • “Bottleneck” analysis in transportation processes employs this technique Stochastic queues - deal with random variables for arrival and service time functions. • Queues are defined by probabilistic metrics such as the expected number of entities in the system, probability of n entities in the system and so on Virginia Polytechnic Institute and State University 7 of 95 Generalized Time Behavior of a Queue State of System Steady-State Transient Behavior Time (hrs) The state of the system goes through two well defined regions of behavior: a) transient and b) steady-state Virginia Polytechnic Institute and State University 8 of 95 Deterministic Queues Deterministic Queues are analogous to a continuous flow of entities passing over a point over time. As Morlok [Morlok, 1976] points out this type of analysis is usually carried out when the number of entities to be simulated is large as this will ensure a better match between the resulting cumulative stepped line representing the state of the system and the continuous approximation line The figure below depicts graphically a deterministic queue characterized by a region where demand exceeds supply for a period of time Virginia Polytechnic Institute and State University 9 of 95 Deterministic Queues (Continuous) Rates Supply Supply Deficit Demand Cumulative Flow Cumulative Demand Lt tin Wt Cumulative Supply tout Virginia Polytechnic Institute and State University Time 10 of 95 Deterministic Queues (Discrete Case) Rates Supply (µ) Supply Deficit Demand (λ) Cumulative Flow Cumulative Demand Cumulative Supply ∆t Virginia Polytechnic Institute and State University Time 11 of 95 Deterministic Queues (Parameters) a) The queue length, L , (i.e., state of the system) corresponds to the maximum ordinate distance between the cumulative demand and supply curves t b) The waiting time, W , denoted by the horizontal distance between the two cumulative curves in the diagram is the individual waiting time of an entity arriving to the queue at time t t in c) The total delay is the area under bounded by these two curves d) The average delay time is the quotient of the total delay and the number of entities processed Virginia Polytechnic Institute and State University 12 of 95 Deterministic Queues e) The average queue length is the quotient of the total delay and the time span of the delay (i.e., the time difference between the end and start of the delay) Assumptions Demand and supply curves are derived derived from known flow rate functions (λ and µ) which of course are functions of time. The diagrams shown represent a simplified scenario arising in many practical situations such as those encountered in traffic engineering (i.e., bottleneck analysis). Virginia Polytechnic Institute and State University 13 of 95 Example : Freeway Bottleneck Analysis A four lane freeway has a total directional demand of 4,000 veh/hr during the morning peak period. One day an accident occurs at the freeway that blocks the right-hand side lane for 30 minutes (at time t=1.0 hours). The capacity per lane is 2,200 veh/hr. a) Find the maximum number of cars queued. b) Find the average delay imposed to all cars during the queue. Virginia Polytechnic Institute and State University 14 of 95 Solution: Plot the Demand and Supply Flows Supply Demand (constant) Virginia Polytechnic Institute and State University 15 of 95 Cumulative Flows = Integral of Flows Cumulative of demand and supply rates of change Virginia Polytechnic Institute and State University 16 of 95 Solutions a) Find the maximum number of cars queued By inspection the maximum number of cars queueing at the bottleneck are 840 passengers. b) Find the average delay imposed to all cars during the queue. Calculate the area under the second curve (in the previous slide) and then divide by the number of cars that were delayed Virginia Polytechnic Institute and State University 17 of 95 Solution : Queue Length and Cumulative Flows Queue length Cumulative Virginia Polytechnic Institute and State University 18 of 95 Some Statistics about the Problem Average arrival rate (aircraft/hr) = 4000 Average capacity (/time) = 3771.4286 Simulation Period (hours) = 5 (hours) Total delay (car-hr) = 1223.7099 Max queue length (cars) = 892.1742 Virginia Polytechnic Institute and State University 19 of 95 Example : Lumped Service Model (Passengers at a Terminal Facility) In the planning program for renovating an airport terminal facility it is important to estimate the requirements for the ground access area. It has been estimated that an hourly capacity of 1500 passengers can be adequately be served with the existing facilities at a medium size regional airport. Due to future expansion plans for the terminal, one third of the ground service area will be closed for 2 hours in order to perform inspection checks to ensure expansion compatibility. A recent passenger count reveals an arrival function as shown below. Virginia Polytechnic Institute and State University 20 of 95 Example Problem (Airport Terminal) λ = 1500 for 0 < t < 1 t in hours λ = 500 for t > 1 where, λ is the arrival function (demand function) and t is the time in hours. Estimate the following parameters: •The maximum queue length, L(t) max •The total delay to passengers, Td •The average length of queue, L •The average waiting time, W •The delay to a passenger arriving 30 minutes hour after the terminal closure Virginia Polytechnic Institute and State University 21 of 95 Example Problem (Airport Terminal) Solution: The demand function has been given explicitly in the statement of the problem. The supply function as stated in the problem are deduced to be, µ = 1000 if t < 2 µ = 1500 if t > 2 Plotting the demand and supply functions might help understanding the problem Virginia Polytechnic Institute and State University 22 of 95 Example Problem (Airport Terminal) Demand and Supply Functions for Example Problem 1: Passengers In 1: 2: 2: Passengers Served 2000.00 1 2 1 2 1: 2: 1000.00 1 2 1: 2: 1 2 0.00 0.00 0.50 1.00 Time Time (hrs) 1.50 12:57 PM Virginia Polytechnic Institute and State University 2.00 7/7/93 23 of 95 Example Problem (Airport Terminal) To find the average queue length (L) during the period of interest, we evaluate the total area under the cumulative curves (to find total delay) Td = 2 [(1/2)(1500-1000)] = 500 passengers-hour Find the maximum number of passengers in the queue, L(t) max, L(t)max = 1500 - 1000 = 500 passengers at time t=1.0 hours Find the average delay to a passenger (W) Virginia Polytechnic Institute and State University 24 of 95 Example Problem (Airport Terminal) TW = ----d Nd = 15 minutes where, Td is the total delay and Nd is the number of passengers that where delayed during the queueing incident. d L = T---tq = 250 passengers where, Td is the total delay and td is the time that the queue lasts. Virginia Polytechnic Institute and State University 25 of 95 Example Problem (Airport Terminal) Now we can find the delay for a passenger entering the terminal 30 minutes after the partial terminal closure occurs. Note that at t = 0.5 hours 750 passengers have entered the terminal before the passenger in question. Thus we need to find the time when the supply function, µ(t), achieves a value of 750 so that the passenger “gets serviced”. This occurs at, µ ( t + ∆ t ) = λ ( t ) = 750 (2.1) therefore ∆t is just 15 minutes (the passenger actually leaves the terminal at a time t+∆t equal to 0.75 hours). This can be shown in the diagram on the next page. Virginia Polytechnic Institute and State University 26 of 95 Example Problem (Airport Terminal) Demand and Supply Functions for Example Problem 1: Passengers In 1: 2: 2: Passengers Served 2000.00 1 2 1 Passenger enters 1: 2: 2 1000.00 1 2 1: 2: 1 Passenger leaves 2 0.00 0.00 0.50 1.00 Time Time (hrs) 1.50 12:57 PM Virginia Polytechnic Institute and State University 2.00 7/7/93 27 of 95 Remarks About Deterministic Queues • • • • Introducing some time variations in the system we can easily grasp the benefit of the simulation Most of the queueing processes at airport terminals are non-steady thus analytic models seldom apply Data typically exist on passenger behaviors over time that can be used to feed these deterministic, non-steady models The capacity function is perhaps the most difficult to wuantify because human performance is affected by the state of the system (i.e., queue length among others) Virginia Polytechnic Institute and State University 28 of 95 Example 2: Deterministic Queueing Model of the Immigration Area at an Airport Let us define a demand function that varies with time representing the typical cycles of operation observed at airport terminals. This demand function, λ ( t ) is: • Deterministic • Observed or predicted • A function of time (a table function) Suppose the capacity of the system, µ ( t ) , is also known and deterministic as shown in the following Matlab code Virginia Polytechnic Institute and State University 29 of 95 Continuous Simulation Example (Rates) 1600 Demand or Cpacity (Entities/time) 1400 Demand - λ(t) 1200 1000 Supply - µ(t) 800 TextEnd 600 400 0 0.5 1 1.5 Time (minutes) 2 Virginia Polytechnic Institute and State University 2.5 3 30 of 95 Plots of Integrals of λ ( t ) and µ ( t ) Entities in Queue 100 80 60 40 20 0 0 TextEnd 0.5 1 1.5 Time 2 2.5 3 1 1.5 Time 2 2.5 3 Total Delay (Entities-time) 100 80 60 40 20 TextEnd 0 0 0.5 Virginia Polytechnic Institute and State University 31 of 95 Matlab Source Code for Deterministic Queueing Model (main file) % Deterministic queueing simulation % T. Trani (Rev. Mar 99) global demand capacity time % Enter demand function as an array of values over time % general demand - capacity relationships % % demand = [70 40 50 60 20 10]; % capacity = [50 50 30 50 40 50]; % time = [0 10 20 30 40 50]; demand = [1500 1000 1200 500 500 500]; capacity = [1200 1200 1000 1000 1200 1200]; time = [0.00 1.00 1.500 1.75 2.00 3.00]; Virginia Polytechnic Institute and State University 32 of 95 % Compute min and maximum values for proper scaling in plots mintime = min(time); maxtime = max(time); maxd = max(demand); maxc = max(capacity); mind = min(demand); minc = min(capacity); scale = round(.2 *(maxc+maxd)/ 2) minplot = min(minc,mind) - scale; maxplot = max(maxc,maxd) + scale; po = [0 0]; passengers to = mintime; tf = maxtime; tspan = [to tf]; % intial number of % where: Virginia Polytechnic Institute and State University 33 of 95 % to is the initial time to solve this equation % tf is the final time % tspan is the time span to solve the simulation [t,p] = ode23('fqueue_2',tspan,po); % Compute statistics Ltmax = max(p(:,1)); tdelay = max(p(:,2)); a_demand = mean(demand); a_capacity = mean(capacity); clc disp([blanks(5),'Deterministic Queueing Model ']) disp(' ') disp(' ') disp([blanks(5),' Average arrival rate (entities/time) = ', num2str(a_demand)]) Virginia Polytechnic Institute and State University 34 of 95 disp([blanks(5),' Average capacity (entities/time) = ', num2str(a_capacity)]) disp([blanks(5),' Simulation Period (time units) = ', num2str(maxtime)]) disp(' ') disp(' ') disp([blanks(5),' Total delay (entities-time) = ', num2str(tdelay)]) disp([blanks(5),' Max queue length (entities) = ', num2str(Ltmax)]) disp(' ') pause % Plot the demand and supply functions plot(time,demand,'b',time,capacity,'k') xlabel('Time (minutes)') ylabel('Demand or Cpacity (Entities/time)') axis([mintime maxtime minplot maxplot]) Virginia Polytechnic Institute and State University 35 of 95 grid pause % Plot the results of the numerical integration procedure subplot(2,1,1) plot(t,p(:,1),'b') xlabel('Time') ylabel('Entities in Queue') grid subplot(2,1,2) plot(t,p(:,2),'k') xlabel('Time') ylabel('Total Delay (Entities-time)') grid Virginia Polytechnic Institute and State University 36 of 95 Matlab Source Code for Deterministic Queueing Model (function file) % Function file to integrate numerically a differential equation % describing a deterministic queueing system function pprime = fqueue_2(t,p) global demand capacity time % Define the rate equations demand_table = interp1(time,demand,t); capacity_table = interp1(time,capacity,t); if (demand_table < capacity_table) & (p > 0) pprime(1) = demand_table - capacity_table; % rate of change in state variable elseif demand_table > capacity_table pprime(1) = demand_table - capacity_table; % rate of change in state variable Virginia Polytechnic Institute and State University 37 of 95 else end pprime(1) = 0.0; % avoids accumulation of entities pprime(2) = p(1); curve over time pprime = pprime'; % integrates the delay Virginia Polytechnic Institute and State University 38 of 95 Output of Deterministic Queueing Model Deterministic Queueing Model Average arrival rate (entities/time) = 866.6667 Average capacity (entities/time) = 1133.3333 Simulation Period (time units) = 3 Total delay (entities-time) = 94.8925 Max queue length (entities) = 89.6247 Virginia Polytechnic Institute and State University 39 of 95 Stochastic Queueing Theory (Nomenclature per Hillier and Lieberman) These models can only be generalized for simple arrival and departure functions since the involvement of complex functions make their analytic solution almost impossible to achieve. The process to be described first is the socalled birth and death process that is completely analogous to the arrival and departure of an entity from the queueing system in hand. Before we try to describe the mathematical equations it is necessary to understand the basic principles of the stochastic queue and its nomenclature. Virginia Polytechnic Institute and State University 40 of 95 Fundamental Elements of a Queueing System Queueing System Queue Entering Customers Service Facility Served Customers Virginia Polytechnic Institute and State University 41 of 95 Nomenclature Queue length = No. of customers waiting for service L(t) = State of the system - customers in queue at time t N(t) = Number of customers in queueing system at time t P(t) = Prob. of exactly n customers are in queueing system at time t s= No. of servers (parallel service) λn = Mean arrival rate µn = Mean service rate for overall system Virginia Polytechnic Institute and State University 42 of 95 Other Definitions in Queueing Systems If λn is constant for all n then (1/λ) it represents the interarrival time. Also, is µn is constant for all n > 1 (constant for each busy server) then µn = m service rate and (1/µ) is the service time (mean). Finally, for a multiserver system sµ is the total service rate and also ρ = l/sµ is the utilization factor. This is the amount of time that the service facility is being used. Virginia Polytechnic Institute and State University 43 of 95 Stochastic Queueing Systems The idea behind the queueing process is to analyze steady-state conditions. Lets define some notation applicable for steady-state conditions, N = No. of customers in queueing system Pn = Prob. of exactly n customers are in queueing system L = Expected no. of customers in queueing system Lq = Queue length (expected) W = Waiting time in system (includes service time) Wq = Waiting time in queue Virginia Polytechnic Institute and State University 44 of 95 There are some basic relationships that have beed derived in standard textbooks in operations research [Hillier and Lieberman, 1991]. Some of these more basic relationships are: L = λW Lq = λWq The analysis of stochastic queueing systems can be easily understood with the use of “Birth-Death” rate diagrams as illustrated in the next figure. Here the transitions of a system are illustrated by the state conditions 0, 1, 2, 3,.. etc. Each state corresponds to a situation where there are n customers in the system. This implies that state 0 means that the system is idle (i,e., no customers), system at state 1 means there is one customer and so forth. Virginia Polytechnic Institute and State University 45 of 95 Rate Diagram for Birth-and-Rate Process λ0 1 0 µ1 λn-1 λ1 2 ....... n n-1 µ2 µn Note: Only possible transitions in the state of the system are shown. Virginia Polytechnic Institute and State University 46 of 95 Stochastic Queueing Systems For a queue to achieve steady-state we require that all rates in equal the rates out or in other words that all transitions out are equal to all the transitions in. This implies that there has to be a balance between entering and leaving entities. Consider state 0. This state can only be reached from state 1 if one departure occurs. The steady state probability of being in state 1 (P1) represents the portion of the time that it would be possible to enter state 0. The mean rate at which this happens is µ1P1. Using the same argument the mean occurrence rate of the leaving incidents must be λ0 P0 to the balance equation, Virginia Polytechnic Institute and State University 47 of 95 Stochastic Queueing Systems µ 1 P 1 = λ 0 P0 For every other state there are two possible transitions. Both into and out of the state. λ 0 P0 = P1 µ1 λ 0 P 0 + µ2 P2 = λ 1 P 1 + µ1 P1 λ 1 P 1 + µ3 P3 = λ 2 P 2 + µ2 P2 λ 2 P 2 + µ4 P4 = λ 3 P 3 + µ3 P3 until, λn-1 Pn-1 + µn+1 Pn+1 = λnµn + µn Pn Virginia Polytechnic Institute and State University 48 of 95 Stochastic Queueing Systems Since we are interested in the probabilities of the system in every state n want to know the Pn's in the process. The idea is to solve these equations in terms of one variable (say P0) as there is one more variable than equations. For every state we have, P1 = λ0 /µ1 P0 P2 = λlλ0 / µ1 µ2 P0 P3 = λ2 λ1 λ0 / µ1 µ2 µ3 P0 Pn+1 = λn ..... λ1 λ0 / µ1 µ2 ...... µn+1 P0 Virginia Polytechnic Institute and State University 49 of 95 Stochastic Queueing Systems Let Cn be defined as, Cn = λn-1 ..... λ1λ0 / µ1 µ2 ...... µn Once this is accomplished we can determine the values of all probabilities since the sum of all have to equate to unity. n ∑P =1 n i=0 n P0 + ∑P n =1 i=1 Virginia Polytechnic Institute and State University 50 of 95 Stochastic Queueing Systems n P0 + ∑C P n 0 =1 i=1 Solving for P0 we have, 1 P 0 = ----------------------n 1+ ∑C n i=1 Now we are in the position to solve for the remaining queue parameters, L the average no. of entities in the system, Lq, the average number of customers in the Virginia Polytechnic Institute and State University 51 of 95 queue, W, the average waiting time in the system and Wq the average waiting time in the queue. Pn = Cn P0 ∞ L= ∑ nP n n=1 ∞ Lq = ∑ (n – s)P n n=s W=L -λ Lq W q = ---λ Virginia Polytechnic Institute and State University 52 of 95 This process can then be repeated for specific queueing scenarios where the number of customers is finite, infinite, etc. and for one or multiple servers. All systems can be derived using “birth-death” rate diagrams. Virginia Polytechnic Institute and State University 53 of 95 Stochastic Queueing Systems Depending on the simplifying assumptions made, queueing systems can be solved analytically. The following section presents equations for the following queueing systems when poisson arrivals and negative exponential service times apply: a) Single server - infinite source (constant λ b) Multiple server - infinite source (constant c) Single server - finite source (constant λ d) Multiple server - finite source (constant Virginia Polytechnic Institute and State University and µ ) λ and µ ) and µ ) λ and µ ) 54 of 95 Stochastic Queueing Systems Nomenclature The idea behind the queueing process is to analyze steady-state conditions. Lets define some notation applicable for steady-state conditions, N = No. of customers in queueing system Pn = Prob. of exactly n customers are in queueing system L = Expected no. of customers in queueing system Lq = Queue length (expected) W = Waiting time in system (includes service time) Wq = Waiting time in queue Virginia Polytechnic Institute and State University 55 of 95 Stochastic Queueing Systems There are some basic relationships that have beed derived in standard textbooks in operations research [Hillier and Lieberman, 1991]. Some of these more basic relationships are: L = λW Lq = λWq Virginia Polytechnic Institute and State University 56 of 95 Stochastic Queueing Systems Single server - infinite source (constant λ and µ ) Assumptions: a) Probability between arrivals is negative exponential with parameter λ n b) Probability between service completions is negative exponential with parameter µ n c) Only one arrival or service occurs at a given time Virginia Polytechnic Institute and State University 57 of 95 Single server - Infinite Source (Constant ρ = λ⁄µ λ and µ ) Utilization factor ∞ 1 P 0 = ---------------------- = ρ n ∞ n=0 n 1+ ρ ∑ ∑ –1 1 = ----------- 1 – ρ –1 = 1–ρ n=1 Pn = ρn P0 = ( 1 – ρ ) ρn λ L = ----------µ–λ for n = 0,1,2,3,..... expected number of entities in the system λ2 L q = -------------------( µ – λ )µ expected no. of entities in the queue Virginia Polytechnic Institute and State University 58 of 95 1 W = ----------µ–λ average waiting time in the queueing system λ W q = -------------------( µ – λ )µ average waiting time in the queue P ( W > t ) = e –µ ( 1 – ρ ) t times probability distribution of waiting (including the service potion in the SF) Virginia Polytechnic Institute and State University 59 of 95 Multiple Server Infinite source (constant λ and µ ) Assumptions: a) Probability between arrivals is negative exponential with parameter λ n b) Probability between service completions is negative exponential with parameter µ n c) Only one arrival or service occurs at a given time Virginia Polytechnic Institute and State University 60 of 95 Multiple Server - Infinite Source (constant ρ = λ ⁄ sµ λ , µ) utilization factor of the facility s–1 1 ( λ ⁄ µ ) n ( λ ⁄ µ )-s -------------------------- - P0 = 1 ⁄ ---------------- + ---------------s! 1 – ( λ ⁄ s µ ) n! n=0 ∑ idle probability Pn = ( λ ⁄ µ )n ---------------- P 0 n! 0≤n≤s ( λ ⁄ µ )n ---------------- P 0 n–s s! s n≥s probability of n entities in the system Virginia Polytechnic Institute and State University 61 of 95 s λ ρ P 0 -- µ L = ----------------------2 + --λ s! ( 1 – ρ ) µ expected number of entities in system s λ ρ P 0 -- µ L q = ----------------------2 s! ( 1 – ρ ) Lq W q = ---λ expected number of entities in queue average waiting time in queue 1 W = L = W q + --λ λ average waiting time in system Finally the probability distribution of waiting times is, Virginia Polytechnic Institute and State University 62 of 95 s λ P 0 -- µ 1 – e –µt( s – 1 – λ ⁄ µ ) 1 + --------------------- -------------------------------- s! ( 1 – ρ ) s – 1 – λ ⁄ µ P ( W > t ) = e –µ t if s–1–λ⁄µ = 0 then use 1 – e –µ t ( s – 1 – λ ⁄ µ ) -------------------------------- = µ t s–1–λ⁄µ Virginia Polytechnic Institute and State University 63 of 95 Single Server - Finite Source (constant λ and µ ) Assumptions: a) Interarrival times have a negative exponential PDF with parameter λ n b) Probability between service completions is negative exponential with parameter µ n c) Only one arrival or service occurs at a given time d) M is the total number of entities to be served (calling population) Virginia Polytechnic Institute and State University 64 of 95 Single Server - Finite Source (constant ρ = λ⁄µ and µ ) utilization factor of the facility M P0 = 1 ⁄ λ ∑ n=0 M! -------------------- ( λ ⁄ µ ) n ( M – n )! idle probability M! P n = -------------------- ( λ ⁄ µ ) n P 0 ( M – n )! for µ+λ L q = M – ------------ ( 1 – P 0 ) λ expected number of entities in n = 1, 2, 3, … M probability of n entities in the system queue Virginia Polytechnic Institute and State University 65 of 95 µ L = M – -- ( 1 – P 0 ) λ expected number of entities in system Lq W q = ---λ W=L -λ average waiting time in queue average waiting time in system where: λ = λ(M – L) average arrival rate Virginia Polytechnic Institute and State University 66 of 95 Multiple Server Cases Finite source (constant λ and µ ) Assumptions: a) Interarrival times have a negative exponential PDF with parameter λ n b) Probability between service completions is negative exponential with parameter µ n c) Only one arrival or service occurs at a given time d) M is the total number of entities to be served (calling population) Virginia Polytechnic Institute and State University 67 of 95 Multiple Server - Finite Source (constant µ) ρ = λ ⁄ µs and utilization factor of the facility s–1 P0 = 1 ⁄ λ ∑ n=0 M! ------------------------- ( λ ⁄ µ ) n + ( M – n )! n! M ∑ n=s M! --------------------------------- ( λ ⁄ µ ) n ( M – n )! s! s n – s idle probability M! ------------------------- ( λ ⁄ µ ) n P 0 ( M – n )! n! Pn = M! --------------------------------- ( λ ⁄ µ ) n P 0 ( M – n )! s! s n – s 0 0≤n≤s if s≤n≤M Virginia Polytechnic Institute and State University n≥M 68 of 95 M Lq = queue ∑ (n – s)P n expected number of entities in n=s M s–1 ∑ ∑ L= nP n = nP n + L q + s 1 – n=0 n=0 s–1 P n n=0 ∑ expected number of entities in system Lq W q = ---λ W=L -λ average waiting time in queue average waiting time in system Virginia Polytechnic Institute and State University 69 of 95 where: λ = λ(M – L) average arrival rate Virginia Polytechnic Institute and State University 70 of 95 Example (3): Level of Service at Security Checkpoints The airport shown in the next figures has two security checkpoints for all passengers boarding aircraft. Each security check point has two x-ray machines. A survey reveals that on the average a passenger takes 45 seconds to go through the system (negative exponential distribution service time). The arrival rate is known to be random (this equates to a Poisson distribution) with a mean arrival rate of one passenger every 25 seconds. In the design year (2010) the demand for services is expected to grow by 60% compared to that today. Virginia Polytechnic Institute and State University 71 of 95 Relevent Operational Questions a) What is the current utilization of the queueing system (i.e., two x-ray machines)? b) What should be the number of x-ray machines for the design year of this terminal (year 2020) if the maximum tolerable waiting time in the queue is 2 minutes? c) What is the expected number of passengers at the checkpoint area on a typical day in the design year (year 2020)? Assume a 60% growth in demand. d) What is the new utilization of the future facility? e) What is the probability that more than 4 passengers wait for service in the design year? Virginia Polytechnic Institute and State University 72 of 95 Airport Terminal Layout Virginia Polytechnic Institute and State University 73 of 95 Security Check Point Layout Queueing System Service Facility Virginia Polytechnic Institute and State University 74 of 95 Security Check Point Solutions a) Utilization of the facility, ρ. Note that this is a multiple server case with infinite source. ρ = λ / (sµ) = 140/(2*80) = 0.90 Other queueing parameters are found using the steadystate equations for a multi-server queueing system with infinite population are: Idle probability = 0.052632 Expected No. of customers in queue (Lq) = 7.6737 Expected No. of customers in system (L) = 9.4737 Average Waiting Time in Queue = 192 s Average Waiting Time in System = 237 s Virginia Polytechnic Institute and State University 75 of 95 b) The solution to this part is done by trail and error (unless you have access to design charts used in queueing models. As a first trial lets assume that the number of xray machines is 3 (s=3). Finding Po, s–1 P0 = ∑ n=0 1 ( λ ⁄ µ )( λ ⁄ µ ) 2 ----------------s -------------------------- ---------------- + s! 1 – ( λ ⁄ s µ ) n! Po = .0097 or less than 1% of the time the facility is idle Find the waiting time in the queue, Wq = 332 s Since this waiting time violates the desired two minute maximum it is suggested that we try a higher number of x-ray machines to expedite service (at the expense of Virginia Polytechnic Institute and State University 76 of 95 cost). The following figure illustrates the sensitivity of Po and Lq as the number of servers is increased. Note that four x-ray machines are needed to provide the desired average waiting time, Wq. Virginia Polytechnic Institute and State University 77 of 95 Sensitivity of Po with S Note the quick variations in Po as S increases. 0.06 Po - Idle Probability 0.05 0.04 0.03 0.02 TextEnd 0.01 0 3 4 5 6 S - No. of Servers 7 Virginia Polytechnic Institute and State University 8 78 of 95 Sensitivity of L with S 25 L - Customers in System 20 15 10 TextEnd 5 0 3 4 5 6 S - No. of Servers 7 Virginia Polytechnic Institute and State University 8 79 of 95 Sensitivity of Lq with S 25 Lq - Customers in Queue 20 15 10 TextEnd 5 0 3 4 5 6 S - No. of Servers 7 Virginia Polytechnic Institute and State University 8 80 of 95 Sensitivity of Wq with S Wq - Waiting Time in the Queue (hr) 0.12 0.1 0.08 0.06 0.04 Waiting time constraint 0.02 0 TextEnd 3 4 5 6 S - No. of Servers 7 8 This analysis demonstrates that 4 x-ray machines are needed to satisfy the 2-minute operational design constraint. Virginia Polytechnic Institute and State University 81 of 95 Sensitivity of W with S Note how fast the waiting time function decreases with S 0.1 0.09 W - Time in the System (hr) 0.08 0.07 0.06 0.05 0.04 0.03 TextEnd 0.02 0.01 0 3 4 5 6 S - No. of Servers 7 Virginia Polytechnic Institute and State University 8 82 of 95 Security Check Point Results c) The expected number of passengers in the system is (with S = 4), s λ ρ P 0 -- µ L = ----------------------2 + --λ s! ( 1 – ρ ) µ L = 4.04 passengers in the system on the average design year day. d) The utilization of the improved facility (i.e., four x-ray machines) is ρ = λ / (sµ) = 230/ (4*80) = 0.72 Virginia Polytechnic Institute and State University 83 of 95 e) The probability that more than four passengers wait for service is just the probability that more than eight passengers are in the queueing system, since four are being served and more than four wait. 8 P(n > 8) = 1 – ∑P n n=0 where, ( λ ⁄ µ )n P n = ---------------- P 0 n! if n≤s ( λ ⁄ µ )n P n = ---------------- P 0 s! s n – s if n>s Virginia Polytechnic Institute and State University 84 of 95 from where, Pn > 8 is 0.0879. Note that this probability is low and therefore the facility seems properly designed to handle the majority of the expected traffic within the two-minute waiting time constraint. Virginia Polytechnic Institute and State University 85 of 95 PDF of Customers in System (L) The PDF below illustrates the stochastic process resulting from poisson arrivals and neg. exponential service times 0.2 0.18 0.16 Probability 0.14 0.12 0.1 0.08 TextEnd 0.06 0.04 0.02 0 0 2 4 6 8 10 12 Number of entities 14 16 Virginia Polytechnic Institute and State University 18 20 86 of 95 Matlab Computer Code % Multi-server queue equations with infinite population % Sc = Number of servers % Lambda = arrival rate % Mu = Service rate per server % Rho = utilization factor % Po = Idle probability % L = Expected no of entities in the system % Lq = Expected no of entities in the queue % nlast - last probability to be computed % Initial conditions S=5; Lambda=3; Mu = 4/3; Virginia Polytechnic Institute and State University 87 of 95 nlast = 10; value computed % last probability Rho=Lambda/(S*Mu); % Find Po Po_inverse=0; sum_den=0; for i=0:S-1 % denominator (den_1) den_1=(Lambda/Mu)^i/fct(i); sum_den=sum_den+den_1; end for the first term in the den_2=(Lambda/Mu)^S/(fct(S)*(1-Rho)); % for the second part of den (den_2) Po_inverse=sum_den+den_2; Po=1/Po_inverse Virginia Polytechnic Institute and State University 88 of 95 % Find probabilities (Pn) Pn(1) = Po; % Initializes the first element of Pn column vector to be Po n(1) = 0; % Vector to keep track of number of entities in system % loop to compute probabilities of n entities in the system for j=1:1:nlast n(j+1) = j; if (j) <= S Pn(j+1) = (Lambda/Mu)^j/fct(j) * Po; else Pn(j+1) = (Lambda/Mu)^j/(fct(S) * Sc^(j-S)) * Po; end end % Queue measures of effectiveness Lq=(Lambda/Mu)^S*Rho*Po/(fct(S)*(1-Rho)^2) Virginia Polytechnic Institute and State University 89 of 95 L=Lq+Lambda/Mu Wq=Lq/Lambda W = L/Lambda plot(n,Pn) xlabel('Number of entities') ylabel('probability') Virginia Polytechnic Institute and State University 90 of 95 Example 4 - Airport Operations Assume IFR conditions to a large hub airport with • • Arrival rates to metering point are 45 aircraft/hr Service times dictated by in-trail separations (120 s headways) Runway 09L-27R 4300 ft. Common Arrival Metering Point Runway 09R-27L Virginia Polytechnic Institute and State University 91 of 95 Some Results of this Simple Model Parameter Numerical Values λ 45 aircraft/hr to arrival metering point µ 30 aircraft per runway per hour Po 0.143 ρ 0.750 L 3.42 aircraft (includes those in service) Wq 2.57 minutes per aircraft W 4.57 minutes per aircraft Virginia Polytechnic Institute and State University 92 of 95 Sensitivity Analysis Lets vary the arrival rate ( λ ) from 20 to 55 per hour and see the effect on the aircraft delay function. 12 Waiting Time (min) 10 8 6 4 2 0 20 25 30 35 40 Arrival Rate (Aircraft/hr) 45 Virginia Polytechnic Institute and State University 50 55 93 of 95 Sensitivity of L with Demand q The following diagram plots the sensitivity of the expected number of aircraft holding vs. the demand function 10 Holding Aircraft 8 6 4 2 0 20 25 30 35 40 Arrival Rate (Aircraft/hr) 45 Virginia Polytechnic Institute and State University 50 55 94 of 95 Example # 5 Seaport Operations • Seaport facility with 4 berths (a beth is an area where ships dock for loading/unloading) • • Arrivals are random with a mean of 2.5 arrivals per day Average service time for a ship is 0.9 days (assume a negative exponential distribution) • Find: • Expected waiting time and total cost of delays per year if the average delay cost is $12,000 per day per ship Virginia Tech (A.A. Trani) 94a Solution (use Stochastic Queueing Model - Infinite Population) Virginia Tech (A.A. Trani) 94b Solution (Seaport Example) Virginia Tech (A.A. Trani) 94c Solution (Seaport Example) Virginia Tech (A.A. Trani) 94d Others Uses of Queueing Models (Facilities Planning) • Queueing models can be used to estimate the life cycle cost of a facility • Using the expected delays we can estimate times when a facility needs to be upgraded • For example, • Suppose the demand function (i.e., number of ships arriving to port) for ships arriving to port increases 10% per year • Determine the year when new berths will be required if the port authority wants to maintain waiting times below 0.5 days. Virginia Tech (A.A. Trani) 94e Calculations for Seaport Example 0 1 2 3 Virginia Tech (A.A. Trani) Time (years) 94f Calculations for Seaport Example Virginia Tech (A.A. Trani) 94g Calculations for Seaport Example Virginia Tech (A.A. Trani) 94h Undiscounted Annual Delay Costs (Lag Solution) $12,000 per hour per ship delay costs Virginia Tech (A.A. Trani) 94i Construction Cost Profile (Seaport) (Lag Solution) $50 Million per Berth Virginia Tech (A.A. Trani) 94j Total Annual Cost (Seaport - Lag Solution) $50 Million per Berth Virginia Tech (A.A. Trani) 94k Lead Solution for Berth Construction Virginia Tech (A.A. Trani) 94l Lead Solution for Berth Construction Virginia Tech (A.A. Trani) 94m Comparing Both Solutions Life Cycle Cost Analysis Virginia Tech (A.A. Trani) 94n Conclusions About Analytic Queueing Models Advantages: • Good traceability of causality between variables • Good only for first order approximations • Easy to implement Disadvantages: • Too simple to analyze small changes in a complex system • Cannot model transient behaviors very well • Large errors are possible because secondary effects are neglected • Limited to cases where PDF has a close form solution Virginia Polytechnic Institute and State University 95 of 95 ...
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This note was uploaded on 12/31/2011 for the course CEE 3604 taught by Professor Katz during the Fall '08 term at Virginia Tech.

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