Measures of waiting line performance

Five important measures

• The average number of customers waiting, either in line or in the system. • The average time customers wait, either in line or in the system. • System utilization, which refers to the percentage of capacity utilized. • The implied cost of a given level of capacity and its related waiting line. • The probability that an arrival will have to wait for service.

System Utilization

(M/---/S):(---/---/ ∞)

Remarks

• The models in our discussions pertain to systems operating under steady-state conditions; that is, they assume the average arrival and service rates are stable.

• The arrival and service rates, represented by 𝜆 and 𝜇, must be in the same units (e.g., customers per hour, customers per minute)

(M=S)

Formulas for rho and r

All infinite-source models require that system utilization be less than 1 (underloaded system).

Formulas following Little’s Law

The relationships are independent of any probability distribution and require no assumptions about which customers arrive or are serviced, or the order in which they are served.

Formula for Lmax

Rule of thumb: if the computed value of Lmax is less than 0.10 above the next lower integer, round down; else round up.

(length of line that will probably not be exceeded for a specified percent of the time)

EXERCISE 1

Remark

Lq is not always given and is model dependent!!!

Additional formulas, specific for (M/M/1):(FCFS/∞/ ∞)

EXERCISE 2

Additional formulas, specific for (M/D/1):(FCFS/∞/ ∞)

Waiting lines are a consequence of random, highly variable arrival and service rates. If a system can reduce or eliminate the variability of either or both, it can shorten waiting lines noticeably. The effect of a constant service time is to cut in half the average number of customers waiting in line:

The average time customers spend waiting in line is also cut in half.

EXERCISE 3

Change of assumption: The ticket agent processes every request with exactly 3-minute duration.

Additional formulas, specific for (M/M/S):(FCFS/∞/ ∞)

Assume:

• Servers all work at the same average rate.

• Customers form a single waiting line (in order to maintain first-come, first-served processing).

Note that in a situation where there are multiple servers, each with a separate line (e.g., a supermarket), each line would be treated as a single-server system.

Additional formulas, specific for (M/M/S):(FCFS/∞/ ∞)

EXERCISE 4

EXERCISE 5