Queuing Models Basic Concepts

QUEUING MODELS Queuing is the analysis of waiting lines It can be used to: –Determine the # checkout stands to have open at a store –Determine the type of line to have at a bank –Determine the seating procedures at a restaurant –Determine the scheduling of patients at a clinic –Determine landing procedures at an airport –Determine the flow through a production process –Determine the # toll booths to have open on a bridge

COMPONENTS OF QUEUING MODELS Arrival Process Waiting in Line Service/Departure Process Queue -- The waiting line itself System -- All customers in the queuing area –Those in the queue –Those being served

The Queue4 customers in the queue The System 7 customers in the system The Queuing Process 1. Customers arrive according to some arrival pattern 2. Customers may have to wait in a queue 3. Customers are served according to some service distribution and depart

ARRIVAL PROCESS Deterministic or Probabilistic (how?) Determined by # customers in system/balking? Single or batch arrivals? Priority or homogeneous customers? Finite or infinite calling population?

THE WAITING LINE One long line or several smaller lines Jockeying allowed? Finite or infinite line length? Customers leave line before service? Single or tandem queues?

THE SERVICE PROCESS Single or multiple servers? All servers serve at same rate? Deterministic of probabilistic (how?) Speed of service depends on line length? FIFO/LIFO or some other service priority?

OBJECTIVE To design systems that optimize some criteria –Maximizing total profit –Minimizing average wait time for customers –Meeting a desired service level

TYPICAL SERVICE MEASURES LL = Average Number of customers in the system L QL Q = Average Number of customers in the queue WW = Average customer time in the system W QW Q = Average customer waiting time in the queue P nP n = Probability n customers are in the system   = Average number of busy servers (utilization rate)

POISSON ARRIVAL PROCESS REQUIRED CONDITIONS –Orderliness at most one customer will arrive in any small time interval of  t –Stationarity for time intervals of equal length, the probability of n arrivals in the interval is constant –Independence the time to the next arrival is independent of when the last arrival occurred

NUMBER OF ARRIVALS IN TIME t Assume = the average number of arrivals per hour (THE ARRIVAL RATE) k arrivals in t hoursPoisson distribution:For a Poisson process, the probability of k arrivals in t hours has the following Poisson distribution:

Time Between Arrivals 1/The average time between arrivals is 1/ exponential distribution:For a Poisson process, the time between arrivals in hours has the following exponential distribution: f(x) = e - t This means: e - t P(next arrival occurs > t hours from now) = e - t 1- e - t P(next arrival occurs within t hours) = 1- e - t

POISSON SERVICE PROCESS REQUIRED CONDITIONS –Orderliness at most one customer will depart in any small time interval of  t –Stationarity for time intervals of equal length, the probability of completing n potential services in the interval is constant –Independence the time to the completion of a service is independent of when it started –IS THIS A GOOD ASSUMPTION?

NUMBER OF POTENTIAL SERVICES IN TIME t Unlike the arrival process, there must be customers in the system to have services Assume  = the average number of potential services per hour (SERVICE RATE) k potential services in t hours Poisson distributionFor a Poisson process, the probability of k potential services in t hours has the following Poisson distribution:

THE SERVICE TIME 1/ The average service time is 1/  exponential distribution:For a Poisson process, the service time has the following exponential distribution: f(x) =  e -  t This means: e -  t P(the service will take t more hours) = e -  t 1- e -  t P(service will be completed in t hours) = 1- e -  t

TRANSIENT vs. STEADY STATE Steady stateSteady state is the condition that exists after the system has been operational for a while and wild fluctuations have been “smoothed out” transient stateUntil steady state occurs the system is in a transient state -- transiting to steady state It is the long run steady state behavior that we will measureIt is the long run steady state behavior that we will measure

CONDITIONS FOR STEADY STATE For any queuing system to be stable the overall arrival rate must be less than the overall potential service rate, i.e. <  –For one server: <  < k  –For k servers with the same service rate: < k  –For k servers with different service rates: <  1 +  2 +  3 + …+  k <  1 +  2 +  3 + …+  k

STEADY STATE PERFORMANCE MEASURES We’ve mentioned these before: LL = Average Number of customers in the system L QL Q = Average Number of customers in the queue WW = Average customer time in the system W QW Q = Average customer waiting time in the queue P nP n = Probability n customers are in the system   = Average number of busy servers

Little’s Laws Little’s Laws relate L to W and L Q to W Q by: LITTLE’S LAWS L = λW L Q = λW Q

Relationship Between the System and the Queue (# in Sys) = (# in queue) + (# being served) Thus, taking expected values of both sides: E(# in Sys) = E(# in queue) + E(# being served) L = L q +  ρ = λ/μIt can be shown that: ρ = λ/μ

Relationship Between the System and Queue Wait Times (Time in Sys) = (Time in queue) + (Service Time) Thus, taking expected values of both sides: E(Time in Sys) = E(Time in queue) + E(Service Time) W = W Q + 1/μ Thus, from last two slides and Little’s Laws, knowing one of L, W, L q and W q allows us to find the other values. Suppose we know L. L Q = L – λ/μ W = L/λ W Q = L Q /λ

CLASSIFICATION OF QUEUING SYSTEMS Queuing systems are typically classified using a three symbol designation: (Arrival Dist.)/(Service Dist.)/(# servers) Designations for Arrival/Service distributions include: –M = Markovian (Poisson process) –D = Deterministic (Constant) –G = General Sometimes the designation is extended to a 4 th or 5 th symbol to indicate Max queue length and # in population

Review Components of a queuing system –Arrivals, Queue, Services Assumptions for Poisson (Markovian) distribution Requirements for Steady State –Overall service rate > Overall arrival rate Steady State Performance Measures –L, L q, W, W q, p n ’s,  Little’s Laws: L = λW and L Q = λW Q System/Queue: L = L Q + λ/μ and W = W Q +1/μ 3-5 component queue classification