Waiting Line Models ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

{Potential Customers} Waiting Line Models Waiting Line System Source Service {Server} Waiting Area Arrival Process Exit {Customers} {Queue} {Potential Customers} ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Doctor’s consultancy room Waiting Line Models Examples of Waiting Line Systems Service System Customer Server Doctor’s consultancy room Patient Doctor Bank Client Clerk Crossing Car Traffic lights Airport Airplane Runway Fire station Fire Emergency unit Telephone exchange Call Switchboard Service station Petrol pump ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

{Potential Customers} Waiting Line Models Waiting Line System Source Service {Server} Waiting Area Arrival Process Exit {Customers} {Queue} {Potential Customers} ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

{Potential Customers} Waiting Line Models Arrival Process Source Infinite – tourists Finite – machines in factory {Potential Customers} ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

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

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

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

Waiting Line Models Service Process Service configurations (type, number and arrangement of service facilities) 1. Single facility Queue Server Exit Arrival ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

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

Waiting Line Models Service Process 2. Multiple, parallel, identical facilities (MULTIPLE queue) Queues Servers Arrival Exit ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Waiting Line Models Service Process 3. Multiple, parallel, but not identical facilities Queues Servers Arrival Exit ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Waiting Line Models Service Process 4. Serial facilities Queue Server Exit Arrival 5. Combination of facilities ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Waiting Line Models Waiting Line Discipline of the queue FCFS (First-Come, First-Served) Service {Server} ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

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

Waiting Line Models Waiting Line Discipline of the queue PRI (PRIority system) Service {Server} ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Waiting Line Models Waiting Line Discipline of the queue SIRO (Selection In Random Order) Service {Server} ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

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

Waiting Line Models Analysis of Waiting Line Models Time characteristics Average waiting time in the queue Average waiting time in the system ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Waiting Line Models Analysis of Waiting Line Models Number of customers Average number of customers in the queue Average number of customers in the system ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Waiting Line Models Analysis of Waiting Line Models Probability characteristics Probability of empty service facility 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 ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Waiting Line Models Classification of Waiting Line Models Kendall’s notation A / B / C / D / E / F 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 ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Standard Single-Server Exponential Model Waiting Line Models Standard Single-Server Exponential Model ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Waiting Line Models Standard Single-Server Exponential Model ( M / M / 1 / FCFS / ∞ / ∞ ) Queue Server Exit Arrival ___________________________________________________________________________ Quantitative Methods of Management  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 ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Waiting Line Models Standard Single-Server Exponential Model Assumptions Infinite source Unlimited length of queue Queue discipline is FCFS ___________________________________________________________________________ Quantitative Methods of Management  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) ___________________________________________________________________________ Quantitative Methods of Management  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 ___________________________________________________________________________ Quantitative Methods of Management  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) ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Waiting Line Models Standard Single-Server Exponential Model Example – Grocery Average waiting time in the system Average waiting time in the queue ___________________________________________________________________________ Quantitative Methods of Management  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 ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Waiting Line Models Standard Single-Server Exponential Model Example – Grocery Probability of finding exactly N customers in the system P(0) 0.280 P(1) 0.202 P(2) 0.145 P(3) 0.105 ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Waiting Line Models Standard Single-Server Exponential Model Example – Grocery Probability that N > n P{N > 0} 0.720 P{N > 1} 0.518 P{N > 2} 0.373 P{N > 3} 0.269 ___________________________________________________________________________ Quantitative Methods of Management  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} 0.890 P{T > 2 min} 0.792 P{T > 3 min} 0.705 P{T > 4 min} 0.627 ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Computer Simulation ___________________________________________________________________________ Quantitative Methods of Management  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 ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry

Computer Simulation 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 ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry