Common Queuing Situations

Waiting-line System

arrivals

size

limited

unlimited

pattern behavior

waiting line

limited

unlimited

queue discipline

service

queuing system design

single-channel

multiple-channel

single-phase

multiphase

service time distribution

constant

negative exponential

Queuing Cost

The three queuing models here all assume:

• Poisson distribution arrivals

• FIFO discipline

• A single-service phase

Model Name Example

A Single-channel Information counter

system at department store

(M/M/1)

Number Number Arrival Serviceof of Rate Time Population Queue

Channels Phases Pattern Pattern Size Discipline

Single Single Poisson Exponential Unlimited FIFO

Model Name Example

B Multichannel Airline ticket

(M/M/S) counter

Number Number Arrival Serviceof of Rate Time Population Queue

Channels Phases Pattern Pattern Size Discipline

Multi- Single Poisson Exponential Unlimited FIFO

channel

Model Name Example

C Constant- Automated car service wash(M/D/1)

Number Number Arrival Serviceof of Rate Time Population Queue

Channels Phases Pattern Pattern Size Discipline

Single Single Poisson Constant Unlimited FIFO

Model Name Example

D Limited Shop with only a population dozen machines

(finite population) that might break

Number Number Arrival Serviceof of Rate Time Population Queue

Channels Phases Pattern Pattern Size Discipline

Single Single Poisson Exponential Limited FIFO

Model A – Single-Channel (M/M/1)

Sonnylye Lai, a repairman, at the SuicidalAccountant Shop, is able to install new keypad on calculators at an average rate of 3 per hour (or about 1 every 20 minutes). Customers seeking this service arrive at the shop on the average of 2 per hours, following a Poisson distribution. They are served on a first-in, first-out basis and come from a very large (almost infinite) population of possible buyers.

Sonnylye Lai, a repairman, at the SuicidalAccountant Shop, is able to install new keypad on calculators at an average rate of 3 per hour (or about 1 every 20 minutes). Customers seeking this service arrive at the shop on the average of 2 per hours, following a Poisson distribution. They are served on a first-in, first-out basis and come from a very large (almost infinite) population of possible buyers.

The owner of the SuicidalAccountant Shop estimates that the cost of customer waiting time, in terms of customer dissatisfaction and lost goodwill, is $10 per hour of time spent waiting in line. The average calculator has a 2/3-hour wait (Wq). There are approximately 16 calculator serviced per day (2 arrivals per hour times 8 working hours per day). The only other major cost that the owner can identify in the queuing situation is the salary of Sonnylyne, the repairman, who earns $7 per hour, or $56 per day. What is the total expected costs?

Model B – Multichannel (M/M/S)

Model Constant-service (M/D/1)

Jeremy Corporation, collects and compacts aluminum cans and glass bottles in New York City. Its truck drivers currently wait an average of 15 minutes before emptying their loads for recycling. The cost of driver and truck time while they are in queues is valued at $60 per hour. A new automated compactor can be purchased to process truckloads at a constant rate of 12 trucks per hour (that is, 5 minutes per truck). Trucks arrive according to a Poisson distribution at an average rate of 8 per hour. If the new compactor is put in use, the cost will be amortized at a rate of $3 per truck unloaded. Evaluate the costs versus benefits of the purchase.