Mathematical approach to the analysis of waiting lines

Transcription

1 Queueing Theory

2 Mathematical approach to the analysis of waiting lines This theory is applicable to a wide range of service operations, including call centers, banks, post offices, restaurants, theme parks, telecommunications systems, and traffic management. In many situations, the customers are not people but orders waiting to be filled, trucks waiting to be unloaded, jobs waiting to be processed, or equipment awaiting repairs. Still other examples include ships waiting to dock, planes waiting to land, and cars waiting at a stop sign.

3 Mathematical approach to the analysis of waiting lines In reality, customers arrive at random intervals rather than at evenly spaced intervals. The key word is average.

4 PROBLEM In the lean philosophy, there are seven wastes: 1. Inventory 2. Overproduction 3. Waiting time adds no value!!! 4. Unnecessary transporting 5. Processing waste 6. Inefficient work methods 7. Product defects

5 PROBLEM Chief among the reasons to minimize queues are the following: 1. The cost to provide waiting space. 2. A possible loss of business should customers leave the line before being served or refuse to wait at all. 3. A possible loss of goodwill. 4. A possible reduction in customer satisfaction. 5. The resulting congestion that may disrupt other business operations and/or customers.

6 THE GOAL There are two basic categories of cost: those associated with customers waiting for service and those associated with capacity. TC = Customer waiting cost + Capacity cost Capacity costs are the costs of maintaining the ability to provide service. Examples include the number of bays at a car wash, the number of checkouts at a supermarket, the number of repair people to handle equipment breakdowns, and the number of lanes on a highway.

7 Unlike the inventory EOQ model, the minimum point on the total cost curve is not usually where the two cost lines intersect.

8 CHARACTERISTICS The main characteristics are 1. Population source. 2. Number of servers (channels). 3. Arrival and service patterns. 4. Queue discipline (order of service).

9 Population Source INFINITE-SOURCE Examples are supermarkets, drugstores, banks, restaurants, theaters, amusement centers, and toll bridges. Theoretically, large numbers of customers from the calling population can request service at any time. FINITE-SOURCE An example is the repairman responsible for a certain number of machines in a company. The potential number of machines that might need repairs at any one time cannot exceed the number of machines assigned to the repairer. Similarly, a nurse may be responsible for answering patient calls for a 10-bed ward, and a secretary may be responsible for taking dictation from three executives.

10 Number of Servers (Channels) It is generally assumed that each channel can handle one customer at a time. A group of servers working together as a team, such as a surgical team, is treated as a single-channel system. We will focus on single phase systems.

11 Arrivals and Service Patterns Waiting lines are a direct result of arrival and service variability. In many instances, the variabilities can be described by theoretical distributions. The most commonly used models assume that arrival and service rates can be described by a Poisson distribution or, equivalently, that the inter-arrival time and service time can be described by a negative exponential distribution.

12 Arrivals and Service Patterns In practice, it is necessary to verify that the assumptions of being Poisson and Exponentially Distributed are met!!! Sometimes this is done by collecting data and using Statistical Tests (e.g., from STAT 101). >>>Report the section Empirical Distributions from Taha<<<

13 Arrivals and Service Patterns Research has shown that being Poisson (or exponentially) distributed are often appropriate for customer arrivals but less likely to be appropriate for service. In situations where the assumptions are not reasonably satisfied, the alternatives would be to (1) search or develop a more suitable model or (2) resort to computer simulation.

14 Arrivals and Service Patterns We will usually assume that customers are patient, that is, that customers enter the waiting line and remain until they are served. Other possibilities are that (1) waiting customers grow impatient and leave the line (reneging); (2) customers switch to another line (jockeying); or (3) upon arriving, customers decide the line is too long and, therefore, do not enter the line (balking). Rejecting if the number of customers in a queue is already at the maximum number allowed, then the customer could be rejected by the system.

15 Queue Discipline Other possibilities: Service in Random Order (SIRO) Last Come, First Served (LCFS)

16 Queue Discipline (order of service) There is first-come service at banks, stores, theaters, restaurants, four-way stop signs, registration lines, and so on. Examples of systems that do not serve on a first-come basis include hospital emergency rooms, rush orders in a factory, and mainframe computer processing of jobs. In these and similar situations, customers do not all represent the same waiting costs; those with the highest costs (e.g., the most seriously ill) are processed first, even though other customers may have arrived earlier.

17 Kendall-Lee-Taha Notation (a/b/c):(d/e/f) a: description of arrivals distribution b: description of service distribution c: number of parallel servers d: queue discipline e: maximum number allowed in the system f: size of the population source

18 Kendall-Lee-Taha Notation For a and b: D is deterministic M is Markovian (Poisson, Exponential) E k is Erlang or Gamma G is General (e.g., normal)

19 Kendall-Lee-Taha Notation For d: FCFS LCFS SIRO GD is general or any type of discipline

20 Kendall-Lee-Taha Notation Example: (M/M/2):(FCFS/100/ )