Chapter 14

Waiting Lines and Queuing Theory Models

A Starting Example

There are on average 12 customers coming to a candy store per hour. A cashier can take care of a customer in 4 minutes on average.

At least how many cashiers should this store hire if it does not want the customers waiting for more than five minutes on average?

Starting example (cont.)

• If the customers came in exactly every five minutes, and service time is 4 minutes exactly per customer:

• If the customers come randomly at the rate of 12 per hour, and the service times are random around 4 minutes:

Waiting Line Models

• Provide an analytical tool for the managers to consider the trade-offs between the customer satisfaction (in terms of customer waiting time) and the service cost, if customer arrivals and/or service times are uncertain (random).

A Queuing System:

• is composed of customers, servers, and waiting lines.

• A customer comes. If a server is idle, the customer can be served immediately, otherwise he/she has to wait in line.

Arrival Patterns

• Random arrival – arrivals follow Poisson distribution.– parameter: arrival rate (number of

customers per unit time)

• Scheduled arrivals -

,

Patterns of Service Time

• Random service time - The length of service time follows the exponential distribution.– parameter: service rate (number of

customers that can be served per unit time)

• Fixed service time -

,

Service Time and Service Rate

• Service rate

= 1 / avg. service time on a customer

Characteristics of a Queuing System:

• Customer population – finite or infinite• Number of lines.• Number of service channels.• Number of service phases - number of

steps to finish a service.• Priority rule - FIFO, LIFO, preemptive, ...• Customer behavior – enter and stay, balk,

renege

Queuing Models in This Chapter

arrival pattern

service time

pattern

number of

serverspopulation

number of

phases

priority rule

customer behavior

M/M/1 random random 1 infinite 1 FIFO

no balk, no

renege

M/M/s random random s infinite 1 FIFO

no balk, no

renege

M/M/1 finite

random random 1 finite 1 FIFO

no balk, no

renege

Performance of a Service System Is Measured by:

• Average queue length (Lq) - average number of customers in the waiting line.

• Average number of customers in the system (L).• Average waiting time in the queue (Wq).• Average staying time in the system (W).• Utilization rate of servers ( ).

• Probability that n customers in the system (Pn).

‘In system’ vs. ‘In queue’

• ‘System’ contains ‘queue’ and service facilities.

• ‘Number of customers in system’ counts customers waiting in queue and customers being served.

• ‘Number of customers in queue’ counts customers waiting in queue only.

• Difference between ‘waiting time in system’ and ‘waiting time in queue’ - ?

Queuing System Calculations

• Use the formulas on p.601 (if doing hand-calculations)

• Use QM for Windows (We use this method!).

Requirements for Managerial Users

• Using the calculation results of QM to

(1) analyze the performance of a service system,

(2) make decisions on capacity such as number of servers to hire.

M/M/m model

• Random arrivals, random service times, m servers.

• Performance of an M/M/m queuing system is determined by arrival rate , service rate , and number of servers m.

• Software QM for Windows calculates the performances of an M/M/S system. (Note: use a same time base for both and .)

Example: Arnold’s Muffler Shop (p.596)

• Time to install a new muffler is random, and on average, the mechanic Reid Blank can install 3 muffler per hour.

• Customer arrivals are random and at the rate of 2 customers per hour on average.

• Evaluate this service system.

Questions about a Service System

• Probability of zero customer in system?

• Utilization of the service capacity?

• Avg. number of customers in system?

• Avg. number of customers in line?

• Average time a customer spends in system?

• A customer’s average waiting time in line?

• In what percent of time is the server idle?

Cost of a Service System

• Total cost

= Total service cost + Total waiting cost

• Total service cost

= (number of servers)·(unit labor cost)

• Total waiting cost =

(1) ·W·(unit waiting cost in system), or

(2) ·Wq· (unit waiting cost in queue).

Muffler Shop (2) p.598

• Waiting cost for the shop is $10 per hour waiting in line.

• The mechanic Reid Blank is paid $7/hour.

• What is the total hourly cost of this system?

• What is the total daily cost of this system?

Muffler Shop (3) p.599

• If Jimmy Smith is hired to replace Reid Blank, then the service rate can be improved to 4 cars per hour, but Jimmy’s hourly salary is $9.

• Evaluate the system with Jimmy Smith.

• Calculate the total daily cost of the system.

• Should Jimmy be hired to replace Reid?

Muffler Shop (4) p.602

• Suppose the shop opens a second garage bay for installing mufflers, and a new mechanic is hired whose salary and service rate are same as Reid Blank.

• Evaluate the new system with two bays and Reid Blank and the new mechanics.

• Calculate the total daily cost of the system.

• Is this a good alternative?

M/D/1 Model

• Random arrivals, fixed service time, one server.

M/D/1 Example: Compactor p.606

• A new compacting machine compacts a truck of recycling cans in 5 minutes constantly. Trucks coming randomly with rate 8 trucks per hour.

• Evaluate this service system.

Compactor (2) p.606

• Cost for a truck waiting in queue is $60 per hour.

• The amortized cost of the new compactor is $3 per truck unloaded.

• Calculate the total cost for a truck unloaded.

• If the current truck waiting time in line is 15 minutes, then should the company purchase the new compactor?

M/M/1 with Finite Population (Source)

• Random arrivals, random service times, one server, finite customer population.

• This model is used if the population is extraordinarily small.

Arrival Rate of a Customer

• In the M/M/1 with finite population model, arrival rate is defined as “arrival rate of a customer”, or “how often a customer comes”. For example:If a customer goes to a barber shop every 15

days, then this customer’s arrival rate is= 1/15= 0.067 per day = 2 per month.

Example: Printers Repair p.608

• A printer breaks down randomly. On average, it breaks down every 20 hours.

• Repair time is random. On average, it takes 2 hours to repair a broken printer.

• Evaluate this printer-service system. (Who is “customer”?)

• Calculate the total cost if printer downtime cost is $120/hour, and the technician is paid $25/hour.