___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models

Service {Server} ___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Source Arrival Process Waiting Area {Queue}Exit Waiting Line System {Potential Customers} {Customers}

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Examples of Waiting Line Systems Service System CustomerServer Doctor’s consultancy room PatientDoctor BankClientClerk CrossingCar Traffic lights AirportAirplaneRunway Fire station Fire Emergency unit Telephone exchange CallSwitchboard Service station Car Petrol pump

Service {Server} ___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Source Arrival Process Waiting Area {Queue}Exit Waiting Line System {Potential Customers} {Customers}

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Source Arrival Process {Potential Customers} Infinite – tourists Finite – machines in factory

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Arrival Process In batches – BUS of tourists Arrivals Individually – patients Scheduled – trams, trains Arrivals Unscheduled – patients

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Arrival Process Time Arrival Arrival Arrival Arrival rate – number of arrivals per time unit (POISSON distribution) Average arrival rate = – average number of arrivals per time unit (mean of POISSON distribution) Average arrival rate = – average number of arrivals per time unit (mean of POISSON distribution)

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Arrival Process Time Arrival ArrivalArrival Average interarrival time = 1/ – average time period beetween arrivals (mean of EXPONENTIAL distribution) Interarrival time – time period between two arrivals (EXPONENTIAL distribution)

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Service Process Service {Server} Service rate – number of customers served per time unit (POISSON distribution) Average service rate =  – average number of customers served per time unit (mean of POISSON distribution) Average service rate =  – average number of customers served per time unit (mean of POISSON distribution)

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Service Process Service {Server} Service time – time customer spends at service facility (EXPONENTIAL distribution) Average service time = 1/  – average time customers spend at service facility (mean of EXPONENTIAL distribution) Average service time = 1/  – average time customers spend at service facility (mean of EXPONENTIAL distribution)

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Service Process  Service configurations (type, number and arrangement of service facilities) 1. Single facility Queue Server ExitArrival

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Service Process 2. Multiple, parallel, identical facilities (SINGLE queue) Queue Servers Arrival Exit

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Service Process 2. Multiple, parallel, identical facilities (MULTIPLE queue) QueuesServers Arrival Exit

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Service Process 3. Multiple, parallel, but not identical facilities Queues Servers Arrival Exit

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Service Process 4. Serial facilities Queue Server ExitArrival Queue Server 5. Combination of facilities

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Waiting Line FCFS (First-Come, First-Served)  Discipline of the queue Service {Server}

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Waiting Line LCFS (Last-Come, First-Served)  Discipline of the queue Service {Server}

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Waiting Line PRI (PRIority system)  Discipline of the queue Service {Server}

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Waiting Line SIRO (Selection In Random Order)  Discipline of the queue Service {Server}

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Analysis of Waiting Line Models  Waiting cost Cost  Service cost (facility cost) - cost of construction - cost of operation - cost of maintenance and repair - other costs (insurance, rental)

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Analysis of Waiting Line Models  Average waiting time in the queue Time characteristics  Average waiting time in the system

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Analysis of Waiting Line Models  Average number of customers in the queue Number of customers  Average number of customers in the system

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Analysis of Waiting Line Models  Probability of empty service facility Probability characteristics  Probability of the service facility being busy  Probability of finding N customers in the system  Probability that N > n  Probability of being in the system longer than time t

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Classification of Waiting Line Models A / B / C / D / E / F A / B / C / D / E / F A / B / C / D / E / F A / B / C / D / E / F Kendall’s notation Probability distribution of interarrival time Probability distribution of service time Number of parallel servers Queue discipline Maximum length of queue Size of customer’s source

___________________________________________________________________________ Operations Research  Jan Fábry Standard Single-Server Exponential Model Waiting Line Models

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Standard Single-Server Exponential Model Queue Server ExitArrival ( M / M / 1 / FCFS / ∞ / ∞ )

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Standard Single-Server Exponential Model Assumptions  Single server  Interarrival times - exponential probability distribution with the mean = 1/λ  Service times - exponential probability distribution with the mean = 1/μ

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Standard Single-Server Exponential Model Assumptions  Infinite source  Unlimited length of queue  Queue discipline is FCFS

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Standard Single-Server Exponential Model Example – Grocery  One shop assistant – serves 25 customers per hour (on the average)  From 8 a.m. to 6 p.m. – 18 customers per hour arrive (on the average)

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Standard Single-Server Exponential Model Example – Grocery  Average arrival rate λ = 18 customers per hour  Average service rate μ = 25 customers per hour

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Standard Single-Server Exponential Model Example – Grocery  Utilization of the system – probability that the server is busy – probability that there is at least 1 customer in the system  Probability of an empty facility (server is idle)

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Standard Single-Server Exponential Model Example – Grocery  Average waiting time in the system  Average waiting time in the queue

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Standard Single-Server Exponential Model Example – Grocery  Average number of customers in the system  Average number of customers in the queue

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Standard Single-Server Exponential Model Example – Grocery  Probability of finding exactly N customers in the system P(0) P(1) P(2) P(3) 0.105

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Standard Single-Server Exponential Model Example – Grocery  Probability that N > n P{N > 0} P{N > 1} P{N > 2} P{N > 3} 0.269

___________________________________________________________________________ Operations Research  Jan Fábry Waiting Line Models Standard Single-Server Exponential Model Example – Grocery  Probability of being in the system longer than time t P{T > 1 min} P{T > 2 min} P{T > 3 min} P{T > 4 min} 0.627

___________________________________________________________________________ Operations Research  Jan Fábry Computer Simulation

___________________________________________________________________________ Operations Research  Jan Fábry Computer Simulation Analytical tools Solution Computer simulation Computer simulation is a special method using computer experiments with the model of a real system

___________________________________________________________________________ Operations Research  Jan Fábry  Entity - object that goes through the model  Resource - agent required by the entity  Event - significant change of the system  Activity - process between two events  Generating of random values  Simulation time  Computer simulation language  Animation Computer Simulation