Chapter 1

Introduction

• “Wait-in-line” is a common phenomenon in everywhere.

• Reason: Demand is more than service.

• “How long must a customer wait?” or “ How many people will form in the line?”

• Queueing theory: answers these questions through mathematical analysis.

1.1 Description of the Queueing Problem

• A queueing system: customers arriving for service, waiting for service, leaving the system.

• A “customer” could be any object waiting for being processed.

• Queueing theory: model queueing systems for predicting their behaviors.

1.2 Characteristics of Queueing Processs

• 6 characteristics: (1) arrival pattern of customers, (2) service pattern of servers, (3) queue discipline, (4) system capacity, (5) number of service channels, and (6) number of service stages.

1.2.1 Arrival pattern of customers

• The process of arrivals is stochastic.

• Interarrival times: a probability distribution (PDF)

• sequential or batch/bulk arrivals (need to know the PDF of the size)

• The reaction upon entering the system – Wait or leave?– Balked: leave for observing the line being too l

ong.– Reneged: leave for waiting too long.– Jockey: switch from line to line.

• Time-dependent (non-stationary) or time-independent (stationary).

1.2.2 Service Patterns

• Service time: a PDF.

• Single or batch (e.g., parallel processing or guided tour)

• State-independent or State-dependent (the service process depends on the number of customers waiting in line).

• Time-dependent (non-stationary) or time-independent (stationary).

• Queue length distribution: the result of arrival and service processes.

1.2.3 Queueing Discipline

• How customers are selected from the queue for service?

• FCFS: first come, first served

• LCFS: last come, first served

• RSS: random selection for service

• PR: priority– Preemptive – Non-preemptive

1.2.4 System Capacity

• finite or infinite queue length

• A finite queue can be viewed as one with forced balking.

1.2.5 Number of Service Channels

• Single or multiple service channels

• Multiple service channels– single queue shared by all channels (e.g., salon)– each channel has its own queue (e.g.,

supermarket)

1.2.6 Stages of Service

• Single or multiple stages (e.g., health exam)

• Multiple stages with recycling or feedback

1.3 Notation

• A/B/X/Y/Z

• A: Interarrival-time PDF

• B: Service-time PDF

• X: # of parallel server channels

• Y: System capacity

• Z: Queue disciplne

• see Table 1.1

1.4 Measuring System Performance

• Effectiveness: Random variables of interest:– Waiting time: system or queue– Accumulation in queue: system or queue– Server idle time (utilization)

• Objectives:– Measuring the effectiveness of a existing syste

m– Design an “optimal” system.– Trade-off of better customer service vs. the exp

ense of providing more service capability.

1.5 Some General Results

: average arrival rate: average service rate

• c: the number of servers

• Traffic intensity: c

• steady state: time goes infinity>1: non-steady state =1: steady only when deterministic arrival

and service <1: steady always