# Waiting Lines and Queuing Theory Models

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Waiting Lines and Queuing Theory Models
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
= 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 servers population number of phases priority rule customer behavior M/M/1 random 1 infinite FIFO no balk, no renege M/M/s s M/M/1 finite finite

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= 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. In M/M/1 with finite population: Total waiting cost ≠ *W*Cw since L ≠ *W. Correct formula is: Total waiting cost = L * Cw. This formula for total waiting cost is correct for all queuing models.

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