Post on 30-Dec-2015
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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 -
,
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 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.