Queuing ModelsQueuing Models

Basic Concepts

QUEUING MODELSQUEUING 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 COMPONENTS OF QUEUING MODELSMODELS

• 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 System7 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 PROCESSARRIVAL 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 LINETHE 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 PROCESSTHE 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?

OBJECTIVEOBJECTIVE

• To design systems that optimize some criteria– Maximizing total profit– Minimizing average wait time for

customers– Meeting a desired service level

TYPICAL SERVICE MEASURESTYPICAL SERVICE MEASURES

• LL = Average Number of customers in the system

• LLQQ = Average Number of customers in the queue

• WW = Average customer time in the system

• WWQQ = Average customer waiting time in the queue

• PPnn = Probability n customers are in the system

= Average number of busy servers (utilization rate)

POISSON ARRIVAL PROCESSPOISSON 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 tNUMBER OF ARRIVALS IN TIME t

• Assume = the average number of arrivals per hour (THE ARRIVAL RATE)

• For a Poisson process, the probability of k k arrivals in t hoursarrivals in t hours has the following Poisson Poisson distribution:distribution:

k!

t)( t)in time arrivalsP(k

k te

Time Between ArrivalsTime Between Arrivals

• The average time between arrivals is 1/1/• For a Poisson process, the time between arrivals in

hours has the following exponential distribution:exponential distribution:

f(x) = f(x) = ee--tt

This means:

P(next arrival occurs > t hours from now) = ee--tt

P(next arrival occurs within t hours) = 1- e1- e--tt

POISSON SERVICE PROCESSPOISSON 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?IS THIS A GOOD ASSUMPTION?

NUMBER OF POTENTIAL NUMBER OF POTENTIAL SERVICES IN TIME tSERVICES 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)

• For a Poisson process, the probability of k k potentialpotential services in t hours services in t hours has the following Poisson distributionPoisson distribution:

t)in time services potentialP(k k!

eμt μtk

THE SERVICE TIMETHE SERVICE TIME• The average service time is 1/1/• For a Poisson process, the service time has

the following exponential distribution:exponential distribution:

f(x) = f(x) = ee--tt

This means: P(the service will take t more hours) = ee--tt

P(service will be completed in t hours) = 1- e1- e--tt

TRANSIENT vs. STEADY STATETRANSIENT 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”

• Until steady state occurs the system is in a transient statetransient state -- transiting to steady state

• It is the long run steady state behavior that It is the long run steady state behavior that we will measurewe will measure

CONDITIONS FOR CONDITIONS FOR STEADY STATESTEADY 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: < < – For k servers with the same service rate: < k < k– For k servers with different service rates:

< < 11 + + 22 + + 33 + …+ + …+ kk

STEADY STATESTEADY STATEPERFORMANCE MEASURESPERFORMANCE MEASURES

• We’ve mentioned these before:

• LL = Average Number of customers in the system

• LLQQ = Average Number of customers in the queue

• WW = Average customer time in the system

• WWQQ = Average customer waiting time in the queue

• PPnn = Probability n customers are in the system

= Average number of busy servers

Little’s LawsLittle’s Laws• Little’s Laws relate L to W and LQ to WQ by:

LITTLE’S LAWSLITTLE’S LAWS

L = λWLQ = λWQ

Relationship Between the System Relationship Between the System and the Queueand 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 = LL = Lqq + +

• It can be shown that: ρρ = = λλ//μμ

Relationship Between the System Relationship Between the System and Queue Wait Timesand 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 = WW = WQQ + 1/ + 1/μμ

• Thus, from last two slides and Little’s Laws, knowing one of L, W, Lq and Wq allows us to find the other values. Suppose we know L.

LQ = L – λ/μW = L/λWQ = LQ/λ

CLASSIFICATION OF QUEUING CLASSIFICATION OF QUEUING SYSTEMSSYSTEMS

• 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 4th or 5th symbol to indicate Max queue length and # in population

ReviewReview• 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, Lq, W, Wq, pn’s,

• Little’s Laws: L = λW and LQ = λWQ

• System/Queue: L = LQ + λ/μ and W = WQ +1/μ

• 3-5 component queue classification