Bottleneck = slowest resource Utilization rate = Throughput Rate (capacity used) / capacity

o Ex. Capacity of cashier = 96 customers per shifto Cashier’s throughput rate = 72 customers per shifto Capacity utilization = 72/96 = .75 (busy 75% of the time)o Utilization rate cannot be greater than 1o As utilization increases (↑)waiting time will rise exponentially faster

Ex. Utilization increases from .60 to .75 and waiting time has increased by 1.5 minutes and if utilization now increases from .75 to .90, the average total time will increase by more than 1.5 minute

WIP inventory: # of units in a system at a point in time stored, waiting, or being processed Throughput Time: Average time a unit stays in the system LITTLES LAW: a rule that links various performance measures

o Throughput Time = WIP/Throughput rate o Keeping WIP fixed, reducing throughput time results in a higher throughput rateo Roller coaster quiz example 50 people in front of you how long do you expect to wait? (The Throuput time of the

ride was 300/hr) 50/300 = 1/6 hr = 10 minutes WIP = Throughput Rate x Throughput Time Inter-arrival time: average time of any two subsequent people waiting in line Square = activity

o Upsidedown Triangle = storage o Diamond = decision

YesNo

Probability Probability that A doesn’t occur = P(not A) = 1 – P(A)Coefficient of Variation = Standard Deviation / Mean

Queuing As the utilization increases, the waiting time and the number of orders in the queue increases exponentiallyPooling the demand (customers) in the one common line improves the performance of the systemQueuing systems are not linear. A small change in the arrival process (service) may result in a very large in waiting time and number of customers that are waiting.

Define a queuing system by -Arrival pattern/Service Pattern/# of servers/Queue capacity • M: Exponential • G: General (Any distribution)

• M/M/2 - Exponential interarrival times, exponential service times, 2 severs• M/G/8 – Exponential interarrival times, general service times, 8 servers

• Poisson and exponential# of customers coming following a Poisson distribution with mean λ, interarrival time is exponentially distributed with mean 1/ λ.

Waiting line

Collect information

Waiting line

Easy call service

Easy call?Waiting

line

Difficult call service

Waiting line

Survey

Notation:Arrival rate: λ (10 cust. per hour)Average interarrival time: ma = 1/ (1/10 hr = 6 min)Service rate: (12 cust. per hour)Average service time: 1/ (1/12 hr = 5 min)

Standard deviation of interarrival time: Sa5 min)Coefficient of variation for arrivals : Ca = Sa /ma

Average service time: ms = 1/Standard deviation of service time: Ss 1 min)Coefficient of variation for services: Cs = Ss /ms

Number of servers: SUtilization rate = λ / S

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