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WAITING LINES

Service Capacity Planning -Waiting Lines

Chapter 18

WAITING LINES

In Short

• Model-1 • Single Channel: Exponential Service Time (M/M/1) • Model-2 • Single Channel: Constant Service Time (M/D/1) • Model-3 • Multi-Channel: Exponential Service Time (M/M/S)

Model 1 is equivalent to “Model 3 with the number of servers = 1.”

WAITING LINES

Basic Facts

Waiting lines in every day in our lives are dynamic, i.e. arrival/service rates change continuously. In this course, we analyze “steady-state” waiting lines, i.e. average arrival rate(λ), average service rate (µ), average number of customers waiting in the line (or queue) (Lq), average number of customers waiting in the system (Ls), average number of customers getting serviced (i.e. ratio of arrival rate to service rate), etc. Steady state values are same all the time! If mean # of customers waiting in the system is 20, then cost of waiting-line system is 20 man-hours per hour. Cost of the system is determined by Ls.

WAITING LINES

Symbols Used

λ is mean arrival rate µ is mean service rate P is probability of either waiting in queue or system L is mean # of customers in “queue” or “system” W is mean waiting time of a customer in queue or system Suffix of P or L or W q for queue s for system Pq, Ps; Lq, Ls; Wq, Ws

WAITING LINES

Formulas Given in the Exam

Table Given in the Exam WAITING LINES

Basic formulas

(You should learn)

WAITING LINES

The following formulas are applicable to both model-1 (single server model) and model-3 (multi-server) model. Lq Average # customers waiting for service i.e. in queue, in line Ls Average # customers in system (waiting in line+being served) Wq Average time a customer is waiting for service i.e. in line/queue Ws Average time a customer spends in the system (i.e. waiting time in queue or line + service time)

λ Ls = Lq + r = Lq + µ

Ws = Wq + 1

Ls = λ Ws

Ls

Lq = λ Wq

Wq = Lq

µ

Ws =

λ

λ

Intuitive formulas...