Queuing or Waiting Line Models

Presentazione sul tema: "Queuing or Waiting Line Models"— Transcript della presentazione:

1 Queuing or Waiting Line Models

2 Introduzione La Teoria delle Code si propone di sviluppare modelli per lo studio dei fenomeni d’attesa in presenza di una domanda di un servizio. Quando la domanda e/o la capacità di erogazione del servizio sono aleatori, si verificano situazioni in cui chi fornisce il servizio non ha la possibilità di soddisfare immediatamente le richieste.

3 Il Sistema e le Componenti
Un sistema coda è un sistema composto da servitori, capaci di fornire un servizio, e da clienti da servire. I clienti che non trovano un servitore libero al loro arrivo si dispongono in coda in aree di attesa (buffer), e vengono serviti con determinate discipline di servizio. La coda è costituita essenzialmente da due processi stocastici: il processo d'arrivo dei clienti e il processo di servizio.

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

5 Coda con un Servitore

6 Coda con On-Off service
Ciclo di carico di una nave fino al posizionamento della successiva di C ore, una fase di carico di V ore e una fase per la partenza e il posizionamento della nuova di R ore. Si assume che R + V = C; i veicoli con i container arrivano al tasso l costante; gli arrivi in un ciclo lC < mV dove m è il tasso di carico.

7 Il Ciclo N L’area grigia è il ritardo totale W l
n’ numero di veicoli serviti in un ciclo n numero di veicoli ritardati l n’ n m R V t C

8 ( ) Osservazioni C λ μ μR 2 1 n' W w - =
Il numero di veicoli ritardati n = R/(l-1 - m-1) n < n’ = lC L’area grigia è W = nR/2 = 1/2lmR2/(l - m) Il ritardo medio a regime per veicolo è ( ) C λ μ μR 2 1 n' W w - = Il ritardo medio dei veicoli ritardati è R/2.

9 Single Channel Queue Queuing System Server Departures Arrivals
Waiting line (queue)

10 Multiple Channel Queue
Queuing System Server 1 Departures Server 2 Arrivals Waiting line (queue) Server 3

11 Structure In general, the arrival of customers into the system is a random event. Frequently the arrival pattern is modeled as a Poisson process. Service time is also usually a random variable. A distribution commonly used to describe service time is the exponential distribution. The most common queue discipline is first come, first served (FCFS).

12 Classification A three part code of the form A/B/s is used to describe various queuing systems. A identifies the arrival distribution B the service (departure) distribution s the number of servers. Used symbols are: M Markov distributions (Poisson/exponential) D Deterministic (constant) G General distribution (with a known mean, variance).

13 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 servers working at identical service rates.

14 Input Characteristics
l average arrival rate 1/l average time between arrivals µ average service rate for each server 1/µ average service time s standard deviation of the service time

15 Poisson Arrival Process
If the average arrival rate is l customers per time unit, then probability of x arrivals per time unit is given by:

16 Example Example: Suppose that customers arrive at the rate of l = 8 customers per hour. The probability that in a given one hour period there will be exactly six arrivals is:

17 Exponential Service Times
If the average service time is 1/µ time units (that is, the average service rate is µ customers per time unit), the probability that the service time will be less than or equal to t time units is:

18 Example If the average service time 1/µ = 6 minute (1/10 hour), then µ = 10 customers per hour. The probability that the service time is less than or equal 5 minutes (1/12 hours) is:

19 Operating Characteristics
P0 = probability the service facility is idle Lq = average number of units in the queue awaiting service L = average number of units in the system Wq = average time a unit spends in the queue awaiting service W = average time a unit spends in the system

20 Analytical Formulas When the queue discipline is FCFS, analytical formulas have been derived for several different queuing models including the following: M/D/1 M/M/1, M/M/k, M/G/1, M/G/k with blocked customers cleared, and 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.

21 M/D/1 System Average number of people or units waiting for service Average time a person or unit spends in the queue Average number of people or units in the system Average time a unit spends in the system ( ) l - m 2 = q L ( ) l - m 2 = q W l m + = q s L m 1 + = q s W

22 M/M/1 System Average number of people or units waiting for service
Average time a person or unit spends in the queue Average number of people or units in the system Average time a unit spends in the system

23 M/M/k System P0 = probability the service facility is idle
Average number of people or units waiting for service Average time a person or unit spends in the queue Average number of people or units in the system Average time a unit spends in the system