[Factory Physics] 8. Variability Basics (2)

• 본 게시글은 Factory Physics(3e, Wallace J. Hopp) 공부 정리글입니다.
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Queuing Theory is a mathematical study of waiting lines, itanalyzes queue behavior, arrival rates, and service rates to optimize performance. Queueing Theory is applicable in scenarios like customer service lines and manufacturing systems. In this blog, we will foucus on the manfuaturing system only. 

 

 

Queues can take various forms, such as single-server or multi-server configurations, each impacting wait times and service efficiency differently. Understanding these variations is crucial for effective queue management and optimizing system performance.

Queues are categorized based on interarrival time, service time distributions, and the number of servers. This classification aids in optimizing queue management strategies for diverse operational scenarios.

 

Notation & Symbols

Queueing notation: A/B/m/b

A: Interarrival time distribution

B: Service time distribution

m: Number of servers in parallel

b: System capacity (if blank, assume infinite capacity)

 

Symbols for A and B 

D: Deterministic (constant) distribution

M: Markovian (Exponential) distribution

G: General distribution (ex. normal, uniform)

 

For example, let's look at the M/M/1 Queue. According to the notation, the M/M/1 Queue is a queue in which interarrival time and service time follow exponential distributions, with a single server.

 

Parameters

Before diving into Queuing Theory, it's essential to understand parameters and performance measures.

 

Performance measures

 

Important relationships for all single-station systems

 

(utilization: probability that the station is busy)

 

 

 

 

 

 

M/M/1 Queue

 

 

 

 

 

 

M/M/m Queue

 

 

When m=1, this expression simplifies to equation, which provides the precise calculation for queue time in the M/M/1 queue. Leveraging this equation, we can derive formulas for CT(M/M/m), WIP(M/M/m), and WIP_q (M/M/m).

 

 

G/G/1 Queue

 

This approximation boasts several advantageous properties. Firstly, it is exact for the M/M/1 queue. Interestingly, it also holds true for the M/G/1 queue, although this fact may not be immediately apparent from our current discussion. Furthermore, it elegantly decomposes into three distinct terms: a dimensionless variability term V, a utilization term U, and a time term T.

 

 

 

G/G/m Queue

 

M/M/1/b Queue

Until now, our analysis has been confined to systems where queue size is unbounded. In these systems, as utilization approaches 100 percent, the average queue (and cycle time) tends to infinity. However, real-world queues are inherently finite due to constraints such as space, time, or operational policies.

When Utilization, u is not 1

 

 

When Utilization, u is 1

 

 

In either case, we can apply Little’s Law.

 

 

 

G/G/1/b Queue

We examine three cases: (1) when the arrival rate is lower than the production rate (u<1), (2) when the arrival rate surpasses the production rate (u>1), and (3) when the arrival and production rates are equal (u=1)

 

when u < 1

 

 

 

Two additional cases exist, but their formulas are notably complex. Interested readers are encouraged to refer to the textbook for further details.