WAITING LINE MODELSBJQP 2023 MANAGEMENT SCIENCE

Overviewo Significant amount of time spent in waiting lines by people, products, etc. o Providing quick service is an important aspect of quality customer service. o The basis of waiting line analysis is the trade-off between the cost of improving service and the costs associated with making customers wait. o Queuing analysis is a probabilistic form of analysis. o The results are referred to as operating characteristics. o Results are used by managers of queuing operations to make decisions.

Elements of Waiting Line Analysiso Waiting lines form because people or things arrive at a service faster than they can be served. o Most operations have sufficient server capacity to handle customers in the long run. o Customers however, do not arrive at a constant rate nor are they served in an equal amount of time. o Waiting lines are continually increasing and decreasing in length.and approach an average rate of customer arrivals and an average service time, in the long run. o Decisions concerning the management of waiting lines are based on these averages for customer arrivals and service times. o They are used in formulas to compute operating characteristics of the system which in turn form the basis of decision making.

Major Elements of Waiting-Line Systems

First come, first served (FCFS) Priority Classification

Waiting lines are commonly found in a wide range of production and service systems that encounter variable arrival rates and service times.

The Single-Server Waiting Line Systemo Components of a waiting line system include arrivals (customers), servers, (cash register/operator), customers in line form a waiting line. o Customers may also: balking, reneging and jockeying o Factors to consider in analysis: o The queue discipline. o The nature of the calling population o The arrival rate o The service rate.

Single-Server Waiting Line System Single-Server Modelo Assumptions of the basic single-server model: o o o o An infinite calling population A first-come, first-served queue discipline Poisson arrival rate Exponential service times

o Symbols: o P = the arrival rate (average number of arrivals/time period) o Q = the service rate (average number served/time period) o Customers must be served faster than they arrive (P < Q) or an infinitely large queue will build up.

Operating CharacteristicsLq = the average number waiting for service L = the average number in the system (i.e., waiting for service or being served) = the system utilization (percentage of time servers are busy serving customers)=

P0 = the probability of zero units in the system V Wq

the average time customers must wait for service

W = the average time customers spend in the system (i.e., waiting for service and service time)

Formulas for Basic Single Server Model

Formulas for Basic Single Server Model (cont d)

Line and Service Symbols for Average Number Waiting, and Average Waiting and Service Times

EXAMPLEA customer service counter is a single-server system. If the arrival rate is 24 customers per hour arrive at checkout counter and 30 customers per hour can be checked out, compute: a) b) c) d) e) f) g) Probability that no customers are in the waiting line system Average number in the waiting line system Average number in the waiting line Average time a customer spends in the total queuing system Average time a customer spends waiting in the queue to be served Probability that the server is busy Probability that the server is idle

Multiple-Channel Model The multiple-channel model is appropriate when these conditions exist:1. A Poisson arrival rate. 2. A negative exponential service time. 3. First-Come, first-served processing order.

1. More than one server.1. An infinite calling population. 2. No upper limit on queue length. 3. The same mean service rate for all servers.

Multiple-Channel Formulas

Multiple-Channel Formulas (cont d)

ExampleThe Student Travel Agency opened 2 counters at DKG 1 to help students purchase bus tickets. Students arrive at the rate of 4 per hour according to a Poisson distribution and each counter spends an average 12 minutes for the transaction. Determine the operating charateristics (P0, L, Lq, W, Wq and Pq) for this system.