Waiting Line Theory Akhid Yulianto, SE, MSc (log)

BankCustomersTellerDeposit etc. Doctor’sPatientDoctorTreatment office Traffic CarsLightControlled intersection passage Assembly linePartsWorkers Assembly Tool cribWorkersClerksCheck out/in tools Situation Arrivals ServersService Process Waiting Line Examples

Structure of a Waiting Line System Queuing theory is the study of waiting lines. Queuing theory is the study of waiting lines. Four characteristics of a queuing system are: Four characteristics of a queuing system are: the manner in which customers arrive the manner in which customers arrive the time required for service the time required for service the priority determining the order of service the priority determining the order of service the number and configuration of servers in the system. the number and configuration of servers in the system.

Level of service Cost Service cost Total waiting line cost Waiting time cost Optimal Waiting Line Costs

Structure of a Waiting Line System Distribution of Arrivals Distribution of Arrivals Generally, the arrival of customers into the system is a random event. Generally, the arrival of customers into the system is a random event. Frequently the arrival pattern is modeled as a Poisson process. Frequently the arrival pattern is modeled as a Poisson process. Distribution of Service Times Distribution of Service Times Service time is also usually a random variable. Service time is also usually a random variable. A distribution commonly used to describe service time is the exponential distribution. A distribution commonly used to describe service time is the exponential distribution.

Poisson Probability x = Tingkat kedatangan x = Tingkat kedatangan λ = rata rata kedatangan per periode λ = rata rata kedatangan per periode e = e =

Eksponential Probability µ =jumlah unit yang di layani per periode µ =jumlah unit yang di layani per periode e = e =

Structure of a Waiting Line System Queue Discipline Queue Discipline Most common queue discipline is first come, first served (FCFS). Most common queue discipline is first come, first served (FCFS). An elevator is an example of last come, first served (LCFS) queue discipline. An elevator is an example of last come, first served (LCFS) queue discipline. Other disciplines assign priorities to the waiting units and then serve the unit with the highest priority first. Other disciplines assign priorities to the waiting units and then serve the unit with the highest priority first.

Structure of a Waiting Line System S1S1S1S1 S1S1S1S1 S1S1S1S1 S1S1S1S1 S2S2S2S2 S2S2S2S2 S3S3S3S3 S3S3S3S3 Customerleaves Customerleaves Customerarrives Customerarrives Waiting line System System

Queuing Systems A three part code of the form A/B/k is used to describe various queuing systems. A three part code of the form A/B/k is used to describe various queuing systems. A identifies the arrival distribution, B the service (departure) distribution and k the number of channels for the system. A identifies the arrival distribution, B the service (departure) distribution and k the number of channels for the system. Symbols used for the arrival and service processes are: M - Markov distributions (Poisson/exponential), D - Deterministic (constant) and G - General distribution (with a known mean and variance). Symbols used for the arrival and service processes are: M - Markov distributions (Poisson/exponential), D - Deterministic (constant) and G - General distribution (with a known mean and variance). For example, M/M/k refers to a system in which arrivals occur according to a Poisson distribution, service times follow an exponential distribution and there are k (sometimes others say s) servers working at identical service rates. For example, M/M/k refers to a system in which arrivals occur according to a Poisson distribution, service times follow an exponential distribution and there are k (sometimes others say s) servers working at identical service rates.

Queuing System Input Characteristics  = the average arrival rate 1/ = the average time between arrivals 1/ = the average time between arrivals µ = the average service rate for each server µ = the average service rate for each server 1/µ = the average service time 1/µ = the average service time  = the standard deviation of the service time  = the standard deviation of the service time

Analytical Formulas For nearly all queuing systems, there is a relationship between the average time a unit spends in the system or queue and the average number of units in the system or queue. For nearly all queuing systems, there is a relationship between the average time a unit spends in the system or queue and the average number of units in the system or queue. These relationships, known as Little's flow equations are: These relationships, known as Little's flow equations are: L = W and L q = W q L = W and L q = W q

Analytical Formulas When the queue discipline is FCFS, analytical formulas have been derived for several different queuing models including the following: When the queue discipline is FCFS, analytical formulas have been derived for several different queuing models including the following: M/M/1 M/M/1 M/M/k M/M/k M/G/1 M/G/1 M/G/k with blocked customers cleared M/G/k with blocked customers cleared M/M/1 with a finite calling population M/M/1 with a finite calling population Analytical formulas are not available for all possible queuing systems. In this event, insights may be gained through a simulation of the system. Analytical formulas are not available for all possible queuing systems. In this event, insights may be gained through a simulation of the system.

M/M/1 Ls = average number of units in the system (waiting and being served) Ls = average number of units in the system (waiting and being served) Ws = average time a unit spends in the system Ws = average time a unit spends in the system Lq = average number of units waiting in the queue Lq = average number of units waiting in the queue Wq = Average time a unit spends waiting in the queue Wq = Average time a unit spends waiting in the queue Utilization factor for the system Utilization factor for the system Probability of 0 units in the system Probability of 0 units in the system Probability of more than k units in the system, where n is the number of units in the system Probability of more than k units in the system, where n is the number of units in the system

M/M/k Queuing System Multiple channels (with one central waiting line) Multiple channels (with one central waiting line) Poisson arrival-rate distribution Poisson arrival-rate distribution Exponential service-time distribution Exponential service-time distribution Unlimited maximum queue length Unlimited maximum queue length Infinite calling population Infinite calling population Examples: Examples: Four-teller transaction counter in bank Four-teller transaction counter in bank Two-clerk returns counter in retail store Two-clerk returns counter in retail store

M/M/S Ls = average number of units in the system (waiting and being served) Ls = average number of units in the system (waiting and being served) Ws = average time a unit spends in the system Ws = average time a unit spends in the system Lq = average number of units waiting in the queue Lq = average number of units waiting in the queue Wq = Average time a unit spends waiting in the queue Wq = Average time a unit spends waiting in the queue Probability of 0 units in the system Probability of 0 units in the system