Basics of Queueing Theory

1. Basics of Queueing Theory Material from Chapter 2 of Simio and Simulation: Modeling, Analysis, Applications as well as external sources

2. Introduction • Most (not all) simulations are of queueing systems, or networks of queues, modeling real systems • “Customers” show up, wait for “service,” get served, maybe go elsewhere, wait again, get served again, etc., maybe leave Basics of Queueing Theory

3. A Single-Server Queueing System Registration • Often interested in output performance measures (metrics), like • Total number of patients going from entry to exit over a fixed time period • Average time in queue (waiting time), not counting service time • Maximum time in queue • Time-average number of parts in queue (area under number-in-queue function, divided by length of time period) • Maximum number of patients in queue over a fixed time period • Average patient time in system • Maximum patient time in system • Utilization of the server (proportion of time busy) • Could be a node in a larger queueing network … Interarrival times, service times random (need to specify probability distributions) Reg. Desk Arriving patients Departing patients Patient being registered Queue Basics of Queueing Theory

4. Urgent-Care Clinic • Random arrivals (no appointments) • Branching probabilities (independent) • Some questions: • How many staff of which type during which time periods? • How big should the waiting room(s) be? • What would be the effect on patient waits if doctors and nurses tended to decrease or increase the time they spend with patients? • What if 10% more patients arrived (arrival rate increased by 10%)? • Should we serve patients in order of their acuity, or just first-in-first-out (FIFO)? Basics of Queueing Theory

5. Why Study Queueing Theory with Simulation? • Terminology, logic similar to many simulation models • In some cases, can derive exact closed-form formulas for output performance metrics, use to verify (debug) simulation models: • Have simulation model of complex system that does not meet queueing-theory assumptions … but it would if we made some (over-)simplifying extra assumptions like exponential distributions for inter-arrival and service times • Modify the simulation model so that it meets these extra assumptions, run it for a very long time … queueing-theoretic results are typically available only for steady state (a.k.a. long run, infinite horizon) • Compare simulation output with queueing-theoretic results … if they (approximately) agree, then confidence in the simulation is improved. • Restore your simulation model back the way it should be! Basics of Queueing Theory

6. Queueing-System Structure and Terminology • Entities (like customers, patients, jobs) arrive, get served either at a single station or at several stations, may wait in queue(s), and may leave the system (if they do, it’s an open system, otherwise if they never leave it’s closed) • Queueing network could consist of several separate queueing stations, each of which is a single- or multiple-server queue • If multiple server, usually assume that a single queue “feeds” all the parallel servers, and that the servers are identical in service speeds: • Queue disciplines – when an entity can leave the queue and start service, which entity gets to be the next one to be served? • First-in, first-out (FIFO), a.k.a. first-come, first-served (FCFS) • Last-in, first-out (LIFO) – a “stack” of physical or logical objects • Priority – Shortest Job First (SJF); or Maximum Value First (MVF) Basics of Queueing Theory

7. Queueing-System Performance Metrics • (Wq) Time in queue (excluding service time); if in a network, either overall (added up for all queueing waits) or at individual stations • (W) Time in system, including time in queue plus service time (again over the network or at a node) • (Lq) Number of entities in queue (a.k.a. queue length), not including any entities in service (again, over the network or at a node) • (L) Number of entities in system, including in queue plus in service (over network or at a node) • (ρ) Utilization of a server, or of a group of parallel identical servers, the time-average number of individual servers in the group who are busy Basics of Queueing Theory

8. Urgent Care Clinic Kelton et al., 2011, p. 21 Basics of Queueing Theory

9. Example Single-server Queueing System Registration Reg. Desk Departing patients Arriving patients Queue Patient being registered Basics of Queueing Theory

10. Patient Arrival/Registration Time Data Basics of Queueing Theory

11. Queuing Systems • Common assumptions: • c servers with a single queue with FIFO ordering • A1, A2, …, An are IID random variables (interarrival times) • l is the arrival rate • S1, S2, …, Sn are IID random variables (service times) • m is the service rate • A’s and S’s are independent • r = l/cm is the utilization • A/B/c/k queuing system (Kendall’s notation) • M/M/1 • M/M/c • M/G/c • GI/G/c Basics of Queueing Theory

12. Queuing Systems • Define the following performance measures: • Wqi is the delay in the queue for the ith customer • Wi = Wqi + Si is the waiting time in the system for the ith customer • Lq(t) is the number of customers in the queue at time t • L(t) is the number of customers in the system at time t (Lq(t) plus the number being served at time t) Steady state average delay in the queue Steady state average waiting time in the system Basics of Queueing Theory

13. M/M/c Queuing Formulae Askin, R. G. and C. R. Standridge, Modeling and Analysis of Manufacturing Systems, John Wiley & Sons, New York, NY, 1993. Basics of Queueing Theory

14. Queueing Networks • Consist of nodes, each of which is a G/G/c station, connected by arcs representing possible entity travel between nodes • Can also have entity arrivals from outside the network, and entities can exit from any node to outside the system • When an entity leaves a node, it can go out on any of the arcs emanating from that node, with arc probabilities summing to 1 • Assume: • All arrival processes from outside haveexponential interarrival times (a.k.a.Poisson processes), and are independentof each other • All service times are independentexponential (so each node is an M/M/c) • All queue capacities are infinite • Utilization (a.k.a. traffic intensity) locally at each node is < 1 Called a Jackson network with these assumptions; much is known about it Basics of Queueing Theory

15. Queueing Networks Kelton et al., 2011, p. 21 Basics of Queueing Theory

16. Queueing Networks (cont.) • Use all of this to analyze each node in a Jackson network as a stand-alone M/M/c queue, using formulae given earlier • Just have to compute Poisson arrival input/output rates using decomposition, superposition of Poisson processes • Let SignIn be the (Poisson) arrival rate into the Sign In station, assume exponential service times throughout: Analyze each node independently as: Sign In: M/M/2, arrival rate SignIn Registration: M/M/1, arrival rate 0.9SignIn Trauma Rooms: M/M/2, arrival rate 0.1SignIn Exam Rooms: M/M/4, arrival rate 0.9SignIn Treatment Rooms: M/M/2, arrival rate (0.9)(0.6)SignIn + 0.1SignIn = 0.64SignIn Basics of Queueing Theory

17. Queueing Network Example 60% STB ~ expo(4.0) min l = 20/hr B A C 40% STA ~ expo(1.875) min STC ~ expo(6.667) min Basics of Queueing Theory

18. Queueing Theory vs. Simulation • Queueing-theoretic results have the advantage of being exact, i.e., no statistical uncertainty/variation • Simulation results have statistical uncertainty/variation, which needs to be acknowledged and appropriately addressed • But queueing theory has its own shortcomings: • Strong assumptions that may be unrealistic, like exponential service times (mode = 0?), making model validity questionable • Nearly always only for steady-state long-run behavior, so don’t address what happens in the short run • Not available for all inter-arrival/service distributions, or (more importantly) for complex systems (Jackson network is simple, restrictive) • Despite output uncertainty, simulation has major advantages: • No restrictions on input distributions, model form, or complexity … so model validity is facilitated • Can address short-term time frames … in fact, steady-state is harder for simulation (long runs, initialization bias) than for queueing theory • Just have to be mindful of proper statistical design/analysis Basics of Queueing Theory