The concept of Hoshin Kanri provides clarity and focus to an organization, by aligning strategic goals through all levels of an organization. It creates an organic flow of information which runs through an organization, where goals run […]

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Push and pull models of manufacturing are two major approaches implemented by organizations. The decision on which approach to use is often based on their business strategy. Push and pull-based models in manufacturing starts at the […]

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A queueing system is a model with the following structure: customers arrive and join a queue to wait for service given by […]

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Today, I’ll briefly explain how to set-up a model in Microsoft Excel to simulate a […]

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There are a few key behavioral responses or reactions to queues, or waiting. […]

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There are 3 types of activities, 2 of which produce waste:

1. Steps that definitely create value.

[…]

The post Travel Time is Non Value Add appeared first on 6sigma.

]]> In general, Travel Time Sucks. Travel Time is Non Value Add. Lucky for us, Queueing Theory helps us put Travel Time in some perspective. You can also view all 40+ articles on Queueing Theory.

There are 3 types of activities, 2 of which produce waste:

1. Steps that definitely create value.
2. Steps that create no value, but are necessary given the current state of the system.
3. Steps that create no value and can be eliminated.

(2) & (3) naturally create wastes, of which there are 7 types:

1. Over-Production: Producing more than is needed, faster than needed or before needed.
2. Wait-time: Idle time that occurs when co-dependent events are not synchronized.
3. Transportation: Any material movement that does not directly support immediate production.
4. Processing: Redundant effort (production or communication) which adds no value to a product or service.
5. Inventory: Any supply in excess of process or demand requirements.
6. Motion: Any movement of people which does not contribute added value to the product or service.
7. Defect: Repair or rework of a product or service to fulfill customer requirements.

It’s important to understand “Value” in terms of the customer. From the custoemer’s perspective, “Value” could be defined in the form of a question:

Which process steps (and associated costs) do our customers not have to bear?

The answer to that question will define the non-value added steps that ought to be eliminated or reduced in order to bring more value to the customer.

Typically, value from the customer’s perspective is manifested in how long something takes to do. For example, at a restaurant, value is created when the right order is processed, delivered, and in a timely way. Time is critical in any service or hard-goods operation.

A Time-Study

Time-study analysis is common in industrial engineering. Conducting a time-study helps to reveal any waste or problems related to the service or product being used. Time-studies are used by industrial engineers, usability analyst, and also by ethnographers to learn about how people use products and how long it takes people to do something. The data gained from a time-study can be invaluable and can help the firm improve their product, service, or overall operation.

A while ago I was involved in a time-study. To begin, you want to identify elements or tasks, and then break those down into manageable pieces. For example, of an element takes less than 4 seconds to do, then you want to group that element with another one. Below is an example of a time-study for a warehousing process:



 

The above time-study has 4 major parts: the start-up, travel, process (locate, reach, grab, put), and travel again. My job was to time a random sample of people conducting this process, then analyze that data for any insights. Below is what we discovered during this process:



 

What initially began as a time-study, later became a discovery for waste — different types. I didn’t include the value-added time above, just the non-value added time. What the Pareto above shows is that travel time is the largest form of waste, but there are others also that can be eliminated. We also discovered that the travel-time was approximated by a normal distribution, which means we can employ the traditional tools that statistics offers us to see if there was an improvement in the process (t-test, chi-square, etc).



 

What we did for this project was to identify ways in which we can reduce travel time — travel time is a necessary, but we were able to reduce the travel time through some engineering of the location of bins, formation of the aisles, the size of the reserve area versus the picking (prime) area, and by switching to smaller batches per order. We also eliminated some of the other waste shown above and reduce the others.

Cost and Benefit Analysis

We calculated the travel time and associated waste to costs the firm about $230,000 per year (for this factory alone — there were 12 others with the same process). We took the projected hours spent in this process multiplied by the waste percentage (see Pareto above) multiplied by the average fully-burdened labor rate to arrive at a cost number. We eliminated 25% of the the waste and associated time-traps in this process, effectively saving ~$57,000. The cost to implement the improvements were trivial. No capital was purchased, just some simple, basic engineering — common-sense stuff was employed.

How This Relates to Queueing Theory

Eliminating waste reduces time that things are in process, allowing for other items to enter the queue. Reducing waste and time-traps helps the servers in the queue complete jobs and allows the jobs to exit the queue, freeing resources so that others can enter.

Conclusion

Travel time is necessary, but some of it can be reduced or completely eliminated. Travel time and some processing time are elements that customers would rather not pay for — it’s a burden that they shouldn’t have to carry in the form of higher prices or defects. We can be proactive in identifying waste in our processes — in any process — and employing process improvement to unlock the value-add that is in our business, resulting in a better customer experience, lower costs, and perhaps a more profitable business.

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]]> https://6sigma.com/travel-time-waste/feed/ 0 https://6sigma.com/call-centers-as-queueing-systems/ https://6sigma.com/call-centers-as-queueing-systems/#respond Fri, 28 Feb 2025 06:02:54 +0000 https://opexlearning.com/resources/228/call-centers-as-queueing-systems It’s clear that a Call Center is a Queue. You can also view all 40+ articles on Queueing Theory.

The flow of calls begins with K trunk lines that connect to the Call Center. There are w ≤ k work stations, or seats, at which N ≤ w agents serve incoming […]

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]]> It’s clear that a Call Center is a Queue. You can also view all 40+ articles on Queueing Theory.

The flow of calls begins with K trunk lines that connect to the Call Center. There are w ≤ k work stations, or seats, at which N ≤ w agents serve incoming calls. An arriving call that finds all k trunk lines occupied (let’s assume there are 8 trunk lines), receives a busy signal and is blocked from entering the system. Else, it is connected to the Call Center and occupies one of the free lines. If fewer than N agents are busy, then the call is routed directly to an agent and is served. But, if there are more calls in the system and all agents are occupied, then the call waits in the queue until an agent serves a call and that call exits the system. Once the call is served, the resources are released — trunk line, agent, work station.

Some calls are considered retrials, because they are not served and attempt to re-enter the system. For some calls, the customer voluntarily abandons the call and does not call back. Also, customers that are served may attempt to re-enter the system; this happens sometimes if there was no first call resolution (for customer service centers) and/or if the customer needed to order something else (for inbound order centers).



Given the simple system above, the number of trunk lines acts as an upper bound (k). The number of agents acts as an upper bound on how many simultaneous calls can be in service simultaneously. A Call Center manager dynamically changes the number of agents on and off the floor based on expected demand and staffing availability.

Call Centers and other operation centers like it are highly complex and dynamic. The model above is a very simple system (read: simplistic). But, it’s easy to see how Call Centers are Queues and all the tools that Queueing Theory makes available to us can be applied to Call Centers.

There is much I have not discussed in regards to Call Centers. There is much I can say, still, about Call Center terminology, mechanics, mathematics, metrics, how to reduce costs at a Call Center, and the applicaion of Erlang C in most Call Centers and how Erlang C isn’t a very good model. I’ll discuss all these at a later time. For now, just bask in the glory of knowing that Call Centers are, indeed, Queues.

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]]> https://6sigma.com/call-centers-as-queueing-systems/feed/ 0 https://6sigma.com/littles-law-for-product-development/ https://6sigma.com/littles-law-for-product-development/#comments Fri, 28 Feb 2025 06:02:53 +0000 https://opexlearning.com/resources/263/littles-law-for-product-development A queueing system is a model with the following structure: customers arrive and join a queue to wait for service given by n servers. After receiving service, the customer exits the system. A fundamental result of queueing theory is little’s law. This article shows how the Product Development Process and Little’s Law are best friends, […]

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]]> A queueing system is a model with the following structure: customers arrive and join a queue to wait for service given by n servers. After receiving service, the customer exits the system. A fundamental result of queueing theory is little’s law. This article shows how the Product Development Process and Little’s Law are best friends, if you let them. You can also view all 40+ articles on Queueing Theory.

Theorem: for a queueing system in steady state, the average length of the queue is equivalent to the average arrival rate multiplied by the average waiting time. in other words,

L = λW

Little’s Law is a fundamental principle in business, mathematics, and has applications to many real-world problems. One of those real-world problems is in product development.

First, a definition:

• WIP/TIP: Work-in-process of Things-in-process. For the purposes of this article, they are synonymous. Being “in-process” means the work or things have entered a state-of-affairs but have not yet exited. The “work” can be anything: materials, components, sales orders, software code, software testing, projects, customer inquiries, checks, phone calls to return reports suppliers to qualify, repair orders, or emails waiting to be answered, etc.

For product development, we can use a transformation of Little’s Law, like the following:

[(Throughput) = (Things-in-Process) / (Average Completion Rate)]

What this equation tells us and what experience has shown time-after-time, is that the number one driver of Product Development Cycle Time are the “things-in-process”. There is no quicker way to reduce the cycle time by which your company can get a product from concept-to-delivery than through first prioritizing all the projects or products and focusing on the ones that make strategic and tactical sense, and killing the lower priority projects.

You might be thinking: “True, but couldn’t we also increase the average completion rate”? You’re right, but the impact of doing that is much lower than reducing the TIP — that is, influencing the average completion rate is rather difficult and is often a function of available resources, scope creep, market demands and changes, etc. Here’s the bottom line: the number one driver for shipping products quicker is by focusing on the important ones and killing the unimportant ones.

Product (Project) Selection Prioritization Matrix

One easy but effective way or prioritizing a list of projects or products is to group-rank them based on variables. Below is an example of a prioritization matrix that I’ve used in the past:



Here are some general steps:

1. List all the projects/products
2. As a group of core stakeholders and decision-makers, agree on a selection criteria, or list of variables on which to judge each item on the list.
3. As a group, score each item against each variable. Use a likert scale that makes sense.
4. Multiply the various scores across to get an overall score.
5. Rank the projects and sort from highest overall score to the lowest. Then, review to see if the ranking makes sense.
6. If done right, then the important initiatives rise to the top and the unimportant ones fall to the bottom.
7. Decide as a group, based on available resources and strategy, where the cut-off score will be. Cut, then discard the items below that score and focus on the ones above that score. I don’t suggest you table or postpone items below the cut-off, because realistically they will be discarded.

Conclusion

To drive throughput, you can influence the size of WIP/TIP or increase the average completion rate. A balanced approach is good, but you’ll gain a higher yield by focusing on reducing WIP/TIP. In order to do that, you must decide as an organization what to focus on — on items that will bring the biggest return to your customer, shareholders, and the company.

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]]> https://6sigma.com/littles-law-for-product-development/feed/ 1 https://6sigma.com/what-is-waste/ https://6sigma.com/what-is-waste/#comments Fri, 28 Feb 2025 06:02:53 +0000 https://opexlearning.com/resources/223/what-is-waste There are 3 types of activities, 2 of which produce waste:
1. Steps that definitely create value.
2. Steps that create no value, but are necessary given the current state of the system.
3. Steps that create no value and can be eliminated.

(2) & (3) naturally create wastes, of which there are 7 types:

[…]

The post What is Waste? appeared first on 6sigma.

]]> There are 3 types of activities, 2 of which produce waste:
1. Steps that definitely create value.
2. Steps that create no value, but are necessary given the current state of the system.
3. Steps that create no value and can be eliminated.

(2) & (3) naturally create wastes, of which there are 7 types:

1. Over-Production: Producing more than is needed, faster than needed or before needed.
2. Wait-time: Idle time that occurs when co-dependent events are not synchronized.
3. Transportation: Any material movement that does not directly support immediate production.
4. Processing: Redundant effort (production or communication) which adds no value to a product or service.
5. Inventory: Any supply in excess of process or demand requirements.
6. Motion: Any movement of people which does not contribute added value to the product or service.
7. Defect: Repair or rework of a product or service to fulfill customer requirements.

It’s important to understand “Value” in terms of the customer. From the custoemer’s perspective, “Value” could be defined in the form of a question:

Which process steps (and associated costs) do our customers not have to bear?

The answer to that question will define the non-value added steps that ought to be eliminated or reduced in order to bring more value to the customer.

Sometime ago, I was involved in evaluating a major process. Software Engineers, Industrial Engineers, and others were involved. We wanted to know how much waste there was in the process and determine how much could be eliminated. Eliminating waste can also be defined as “how much value could we unlock” or “how much burden do our customers *not* have to bear?”, as defined above. It was a fun excercise and it was quite revealing. Below is what we produced:



Above is the as-is, end-to-end map for the process that we evaluated. On the left-hand side are the activities that we determined to be value-added. On the right-hand side are the wasteful activities that could be eliminated. If the customer were to have her say, I’m sure she would prefer not to bear the costs and burden of the righ-hand side activities. Eliminating non-value added activities would unlock tremendous value and bring satisfaction and cost savings to the customer.

The above example and excercise above was for a manufacturing-type activity. But, the same worldview and approach could be taken for any process or service. In software, for example, there is a lot of waste involved in producing software, much of which could be eliminated.

Today, take a good look at the world around you. What waste, burden, and associated costs do you and your customers have to put up with? What can you do to help reduce the customer’s burden?

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]]> https://6sigma.com/what-is-waste/feed/ 1 https://6sigma.com/recognizing-constraints-bottlenecks/ https://6sigma.com/recognizing-constraints-bottlenecks/#comments Fri, 28 Feb 2025 06:02:19 +0000 https://opexlearning.com/resources/247/recognizing-constraints-bottlenecks How to Recognize a Bottleneck or Constraint in Your System is no easy task. You can also view all 40+ articles on Queueing Theory.

All dynamic systems – online or offline, such as restaurant operations or an ecommerce store – consists of discrete and dependent processes. […]

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]]> How to Recognize a Bottleneck or Constraint in Your System is no easy task. You can also view all 40+ articles on Queueing Theory.

All dynamic systems – online or offline, such as restaurant operations or an ecommerce store – consists of discrete and dependent processes. Each step in the system has a finite capacity. When one step is feeding more than what the next step can handle, you’ll have yourself a constraint. Constraints or Bottlenecks aren’t bad, they’re expected and are found in any process. The key is recognizing and then managing the constraint.

Recognizing Your Constraints

Imagine the following generic process:

It doesn’t even matter what IPH stands for – just look at the raw outputs, because that is what will “feed” the next dependent process step. Do you see the constraint?



The Role of a Bottleneck

1. Bottlenecks determines the throughput of a system.
2. An increase in the bottleneck rates is the only way to increase throughput.
3. All other process steps should be slaves to the bottleneck.
4. It’s okay to take resources from a non-bottleneck if it will help the bottleneck.

Managing Constraints & Bottleneck Principles

1. Bottlenecks should never be idle; to lose time on a bottleneck, is to lose throughput.
2. Never let a bottleneck run out of work. It’s okay to build inventory in front of a bottleneck.
3. Increase productivity rates (offline and online processes) by reducing down-time, change-over time, and off-task time.
4. Reduce defects by having Quality Assurance and Quality Control in front of a bottleneck, not after.
5. Focus all improvements on the bottleneck.

In any offline or online process, there will be contraints. It’s important that you identify the contstraint, then manage it; once you manage it, it’s important to remember that bottlenecks move. When this happens, follow the above steps again to identify, then manage your bottlenecks.

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]]> https://6sigma.com/recognizing-constraints-bottlenecks/feed/ 1 https://6sigma.com/psychology-of-queueing-build-a-bear-workshop/ https://6sigma.com/psychology-of-queueing-build-a-bear-workshop/#comments Fri, 28 Feb 2025 06:02:18 +0000 https://opexlearning.com/resources/246/psychology-of-queueing-build-a-bear-workshop Waiting Line Management at Build a Bear Workshop is interesting. In addition to this article You can also view all 40+ articles on Queueing Theory.

Last Saturday, we celebrated my daughter’s birthday. For her birthday, we took her and 13 of her friends to Build-A-Bear Workshop, which is a business that […]

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]]> Waiting Line Management at Build a Bear Workshop is interesting. In addition to this article You can also view all 40+ articles on Queueing Theory.

Last Saturday, we celebrated my daughter’s birthday. For her birthday, we took her and 13 of her friends to Build-A-Bear Workshop, which is a business that where you can:

1. Choose your Bear
2. Create a voice for your Bear
3. Stuff the Bear
4. Stitch the Bear
5. Fluff the Bear
6. Dress the Bear
7. Name the Bear
8. Then go home with a nice big bear house box



Needless to say, my daughter and all her friends loved it. This business is a mix of merchandising, collectibles, and entertainment — a great business; an entire experience that really satisfies their target customer.

Our group had a Bear Guide, an employee that helped each of the girls through each step. At step 1, there was no bottleneck because there were several bins where the girls could pick out their bear. At Step 2, we chose to skip this one because it was an upsell. At Step 3, there was a fluff machine with 2 available hoses to fluff the bears with. This was the bottleneck in the process and with 13 girls at a birthday party, you wouldn’t want them to wait. Our Bear Guide was great — during this step, she had the girls play a game while she stuffed the bears and stitched them. Little do the girls know, that our guide was employing a tactic in Queueing Psychology — the waiting feels less of a burden if you’re engaged in something interesting or entertaining, while you’re waiting for the real reason for being at the point of service. This was a great demonostration of an effective use of Queueing Psychology.

The rest of the process went well and smoothly. All throughout the process there is singing, dancing, and it’s just fun for the kids.

Out of curiousity, I asked the manager of the store how the store is doing financially. I asked him, on average, how many bears are produced and the average price of the bear. He said:

“About 300 bears are produced per day during the week at about $30/bear. On the weekend, that number goes to about 500 bears per day.”

WoW! This means that this store brings in at least $60,000/week. Very nice.

As a review, below are the steps to better manage the Psychology of Queues:

1. Unoccupied time feels longer than occupied time.
2. Process-waits feel longer than in-process waits.
3. Anxiety makes waits seem longer.
4. Uncertain waits seem longer than known, finite waits.
5. Unfair waits are longer than equitable waits.
6. The more valuable the service, the longer the customer is willing to wait.
7. Solo waits feel longer than group waits.



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]]> https://6sigma.com/psychology-of-queueing-build-a-bear-workshop/feed/ 1 https://6sigma.com/psychology-of-queueing-disneyland/ https://6sigma.com/psychology-of-queueing-disneyland/#comments Fri, 28 Feb 2025 06:02:11 +0000 https://opexlearning.com/resources/372/psychology-of-queueing-disneyland I went on vacation last week to Disneyland.  We had a lot of fun.  It was also a time to learn how organizations like Disneyland deal with queueing challenges, especially with systems under high stress and load.  In this post, I want to cover the Psychology of […]

The post Wait Time Psychology at Disneyland appeared first on 6sigma.

]]> I went on vacation last week to Disneyland.  We had a lot of fun.  It was also a time to learn how organizations like Disneyland deal with queueing challenges, especially with systems under high stress and load.  In this post, I want to cover the Psychology of Queueing and how Disneyland satisfies several of the positive properties. You can also view all 40+ articles on Queueing Theory.

There are a few key behavioral responses or reactions to queues, or waiting.

Wait Time Psychology Principles

Below are the propositions:

1. Unoccupied time feels longer than occupied time.
2. Process-waits feel longer than in-process waits.
3. Anxiety makes waits seem longer.
4. Uncertain waits seem longer than known, finite waits.
5. Unfair waits are longer than equitable waits.
6. The more valuable the service, the longer the customer is willing to wait.
7. Solo waits feel longer than group waits.

In almost every ride at Disneyland, they provide the following metric at the beginning of the ride:



The message above signals to the customer approximately how long she or he is expected to wait.  This strategy satisfies property 4 of the Psychology of Queueing — it is no longer an uncertain wait, but a finite (albeit seemingly long) wait.  This helps to manage the customer’s expectations or helps the customer decide whether he or she is willing to wait or willing to move on to a different ride.  This visual display is a simple, yet powerful move that helps the customer.

Now, to arrive at the average wait time above, requires some understanding of the physics of queueing.  The physics of queueing that helps us approximate the average wait time is below:

1. λ = Arrival Rate, or more specific, the time between arrivals.  For most queues, we can assume that the arrival distribution can be approximated by a Poisson distribution; which means that the time between arrivals are not deterministic, but random.
2. μ = Service Rate, or more specific the time for a arrival to be serviced.

So, we get the following to conclude the average wait time:

T_w = (λ / μ(μ – λ))

The calculation is simple, yet powerful.  Couple the physics of queueing with a positive approach in the psychology of queueing, then you’ll better manage the expectations of your customers, better forecast the load and requirements of a system, and better predict requirements on labor or resources.

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]]> https://6sigma.com/psychology-of-queueing-disneyland/feed/ 6 https://6sigma.com/queueing-disneyland-and-fastpass/ https://6sigma.com/queueing-disneyland-and-fastpass/#comments Fri, 28 Feb 2025 06:02:11 +0000 https://opexlearning.com/resources/374/queueing-disneyland-and-fastpass In a previous article, I discussed the Psychology of Queueing and my experience at Disneyland while on vacation with my family.   In this post, I want to briefly talk about FastPass at Disneyland, in the context of Queueing.

FastPass is a feature that Disneyland offers […]

The post Disneyland Wait Times and Queueing Theory and Impact of Fast Pass appeared first on 6sigma.

]]> In a previous article, I discussed the Psychology of Queueing and my experience at Disneyland while on vacation with my family.   In this post, I want to briefly talk about FastPass at Disneyland, in the context of Queueing.

FastPass is a feature that Disneyland offers its customers, wherein a customer is invited to obtain a pass, redeemable only at a certain time and ends at a certain time.  The idea is that during those times, theoretically, the lines will be shorter and the wait time will be shorter also.  Below is a picture of what that looks like at Disneyland.

As you can see from the picture above, the customers are invited to return between 4:25PM to 5:25PM, with the idea that the wait time will be shorter.  Disneyland calls this offering FastPass.


As I thought more about FastPass, I think it is a simple Queueing calculation; If I’m thinking about this correctly, then, we need to know the following items:

1. λ = Arrival Rate, or more specific, the time between arrivals.  For most queues, we can assume that the arrival distribution can be approximated by a Poisson distribution; which means that the time between arrivals are not deterministic, but random.
2. μ = Service Rate, or more specific the time for a arrival to be serviced.

Then, Disneyland probably has historical data on when this particular ride is least congested, and I’m assuming it’s between the hours of 4:25PM and 5:25PM.  Disneyland, then, can calculate, given historical congestion data and the Arrival Rate and the Service Rate, the probability of N arrivals during this 60 minute period:

(P_n = (λT)ⁿ/n!)e^-λT

Where T is period T, in our Disneyland example, it is 60 minutes.

Let’s assume that Arrival Rate = 15, and we want to know the probability of 4 arrivals during this 60 minute period.  Then, we get the following:

(P₄ = (15*1.0)⁴/4!)e^-15*1.0

The result of the equation above would give a % of probability of 4 arrivals during a 60 minute period.

Does FastPass Work?

I’m not sure.  Our family went on plenty of rides and, on some, I saw noticed that FastPass seemed to work, whereas on others, FastPass was no faster than the regular line.

Like most things, the equations we come up with are clean and neat.  But, when they meet everyday life and empirical data, things get a little muddy and much more complex.

Enough Pontificating

We had a great vacation at Disneyland.  We’re already planning our next vacation — we’re thinking Hawaii.  I can’t wait.

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]]> https://6sigma.com/queueing-disneyland-and-fastpass/feed/ 7 https://6sigma.com/youtubes-queueing-properties/ https://6sigma.com/youtubes-queueing-properties/#comments Fri, 28 Feb 2025 06:01:43 +0000 https://opexlearning.com/resources/349/youtubes-queueing-properties YouTube Upload Processing has many intriguing queueing properties.  This article will primarily look at the mpeg-to-swf conversion and study out the queueing properties of that process. You can also view all 40+ articles on Queueing Theory.

I’ll show the basic process of how to upload a video on Youtube, explain the […]

The post Youtube Upload Processing and Queueing Theory appeared first on 6sigma.

]]> YouTube Upload Processing has many intriguing queueing properties.  This article will primarily look at the mpeg-to-swf conversion and study out the queueing properties of that process. You can also view all 40+ articles on Queueing Theory.

I’ll show the basic process of how to upload a video on Youtube, explain the Queueing mechanics that goes on behind-the-scenes, propose a few books in case you’re interested in Queueing Theory applications, and then finally show a video tutorial on an introduction to Queueing Theory and Models.

I make many assumptions in this article, so feel free to poke holes in what I’m arguing.

Assumptions and Data

Below is a very simple, high-level process map of the steps to upload a video on YouTube:



In summary, the user does the following:
1. User uploads video
2. User waits while video is uploaded
3. Upload is completed
4. User waits for video to be converted from MPEG to SWF
5. Conversion from MPEG to SWF completes

Based on a small sample of 20 uploaded videos of an average file size of 3.5 MB, I calculate the mean for uploads to be ~180 seconds per MB and ~240 seconds per MB for the conversion.

Based on YouTube’s own disclosure, we also know that there are on average 65,000 video uploads per day.  We do not know the average file size.

Queueing Properties

Important items to note when studying the queueing properties of a system are the following:

1. λ = Arrival Rate, or more specific, the time between arrivals.  For most queues, we can assume that the arrival distribution can be approximated by a Poisson distribution; which means that the time between arrivals are not deterministic, but random.
2. μ = Service Rate, or more specific the time for a arrival to be serviced.

A poisson distribution typically looks skewed to the left or to the right — that is because the mean and the standard deviation is the same.  Here’s a standard picture of a poisson for server utilization:



What we see above is that as there are more simultaneous connections, there is a subsequent arrival rate batching — represented by the poisson curve above.

Given the data and notation, we can now attempt to better understand the queueing properties of the MPEG-to-SWF conversion.  Remember: the data I have assumes several things and, is most likely, completely off the mark.  But, it’s an attempt and, if anything, it’s fun to try.

Presuppostions Redux

So, 65,000 video uploads on a 12 hour day, gives us the following:

λ = 65,000 / 720 minutes = (90 / minute)
μ = 3.5 * 240 seconds = 840 seconds; 840 seconds / 60 = (14 conversion / minute)

Arguably, 14 conversion / minute is very low.  Let’s just assume that YouTube average service rate is 200 conversions / minute.   Given that, we can now learn about the queueing properties, which I describe below.

Average Number of Videos Waiting to be Converted

The equation to learn about the average number of files in the conversion process is the following:

C_w = (λ² / μ(μ – λ))

So,

C_{w =} [(8100) / (200(200 – 90)] = .36

So, given the assumptions above, not even 1 video is waiting to be converted from MPEG-to-SWF.

Average Number of Videos in the Conversion Process

C_s = (μ – λ)

So,

C_{s =} [(90) / (200 – 90)] = .81

This means, given the assumptions above, that an any point in time, there is 1 video in the conversion system.

Average Time Spent Waiting

T_w = (λ / μ(μ – λ))

So, we get:

T_{w =} [(90) / (200(200 – 90))] = .004

This means as videos enter the conversion process, there is hardly any waiting — they are served almost immediately.

Average Time Spent in the System

T_s = (1 / (μ – λ))

So, we get,

T_{s =} (1 / (200 – 90)) = .009

This means, then, that as videos are uploaded and enter the conversion queue, they are served almost immediately, without any waiting.

Weaknesses in Analysis

Okay, the numbers above are pretty much pulled out of the clouds.  But, if we had real data, then you could just plug them into the equations above.  My guess is, though, that even if we had real numbers, the results would be really close to what I show above.  Why?  Well, for each input into the YouTube system, one could argue that it has very little impact on resources — this is a common property in telephony and in server modeling.  I see the same thing going on with YouTube.  The biggest challenge for YouTube is not computing resources, but storage capacity.

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