Aircraft systems

Waiting line: use In manufacturing and service organizations to understand and arrive solutions To eliminate or minimize them.

Waiting lines tend to form if arrival and service patterns are highly variable because the Variability creates temporary imbalances of supply and demand.

Structure of waiting Line problems: An input, or customer population that generates potential Customers. A waiting line of Customers. The service facility, Consisting of a person (or crew), a machine (or group of machines), or both Necessary to perform the service for the customer. A priority rule, which selects the next customer to be served by The service facility

Customer population

The source of input: Finite or infinite source

Customers already in line from a finite source reduce the Chance of new arrivals

Customers already in line from an infinite source do not Affect the probability of another arrival

Customers are patient Or impatient: Patient customers wait until served

Impatient customers either balk (i.E. Do not enter the System) or renege (i.E. Leave the system before being served)

The service system:

Number of lines: Single, Multiple

Arrangement of Service facilities:

Single-channel, single-phase. Single-channel, multiple-phase. Multiple-channel, single-phase. Multiple-channel, multiple-phase. Mixed Arrangement

Priority rule: First-come, first-served (FCFS)—used by most service systems

Other rules:

Earliest due date (EDD)

Shortest processing time (SPT)

Preemptive discipline—allows A higher priority customer to interrupt the service of another customer or be Served ahead of another who would have been served first

Probability Distribution: The sources of variation in waiting-line problems come from The random arrivals of customers and the variation of service times

Arrival distribution: Customer arrivals can often be described by the Poisson distribution with Mean  = T and variance also = T

Interarrival times are the average time between arrivals

Service time Distribution: Service time distribution can be described by an exponential Distribution with mean  = 1/  and variance  = (1/ )2

Single server model: Single-server, Single line of customers, and only one phase

Assumptions:

Customer population is infinite and patient

Customers arrive according to a Poisson distribution, with a Mean arrival rate of 

Service distribution is exponential with a mean service rate Of 

Mean service rate exceeds mean arrival rate

Customers are served FCFS

The length of the waiting line is unlimited

SINGLE SERVER MODEL FORMULA: Both the average waiting time in the system (W) and the average Time spent waiting in line (Wq) are expressed in hours. To convert the results To minutes, simply multiply by 60 minutes/ hour. For example, W = 0.20(60) = 12 Minutes, and Wq = 0.1714(60) = 10.28 minutes.

Multiple server model

Service system has Only one phase, multiple-channels

Assumptions (in Addition to single-server model)

There are s identical servers

The service distribution for each server is exponential

The mean service time is 1/

s should always exceed 

Little’s law

Relates the number of customers in a waiting-line system to The arrival rate and the waiting time of customers

 L = W  or Lq = Wq

L = average number of customers in the system

 = average arrival rate

W  = average time a Customer spends in the system

Service; estimate W. Average time in facility = W= L customers/(A customer/hour)

Manufacturing: Estimate The average work-in-process L

Work-in-process = L = W