Little’s Law Is Big For Startups – TechCrunch

This post is written by Matt Oguz at TechCrunch and the link to the original article is here.

Traffic, traction, growth. We all know that these terms are prerequisites to success. As we launch our startups we hope for initial customer acceptance, which would lead to traffic, traction and growth (TTG). In some cases, we’re willing to pay for traffic. In most other cases, we work around the clock to ignite organic TTG.

When we read about the successful co-founders of a Yelp, Pinterest, or WhatsApp, we find ourselves inspired by their drive and intellect, but we often leave wondering what it really was that gave these startups the astronomical TTG that we all want. There’s certainly no shortage of ideas and opinions about how one startup achieved success, but as analytical founders, the prescribed path from “good to great” often does not satisfy us. We crave more mathematical guidance.

One discipline to turn to in order to understand the underlying mechanics of business is operations research (OR).

OR principles not only guide us to optimize and run our businesses smoothly but also provide us with statistical analysis of underlying business concepts via modeling and simulation. One of the most interesting studies in OR which provides relevant guidance to today’s applications is queuing theory. And inside queuing theory, Little’s law is a hidden gem that gives us profound hints on where to focus to achieve superior traffic, traction and growth.

Queuing theory in its simplest terms tackles problems within the context of the following flow in a store:

Arrival –> Service–> Departure

In a queuing system, there are items that arrive at some rate to the system. Then they depart. An item can be a customer or inventory. When we think about it, this is exactly what we have on a website or app. Visitors arrive, they stick around for a while, then they leave. The most valuable company is the one with the most visitors that stay the longest.

Little’s Law says that, under steady state conditions, the average number of items in a queuing system equals the average rate at which items arrive multiplied by the average time that an item spends in the system.


L =average number of items in the queuing system,

W = average waiting time in the system for an item, and

λ =average number of items arriving per unit time, the law states the following:


“The long-term average number of customers in a stable system is equal to the long-term effective arrival rate multiplied by the average time a customer spends in the store.”

This statement sounds trivial. Its magic, however, lies in the simplicity that the relationship is not influenced by the service distribution, service order or anything else. It’s not influenced by the color of the site, the distribution of the content or the price of the product. The only thing that matters is how fast the visitors are coming and how long they’re staying. Everything else is secondary. Little’s law doesn’t only apply to queues in physical stores; it applies to networks and to any system where there’s a flow of items.

To examine a real-life situation, it’s safe to claim that Google, as a search engine, has the highest arrival rate of visitors, namely λ. But the visitors don’t stick around much. They quickly click through to another site via organic or paid links. Then they come back later for another search only to leave quickly. Google has done a phenomenal job at building up that arrival rate that made the company what it is today. But take a look at the acquisitions, research or any other top initiative at Google, and you’ll easily see that all of them target the second part of Little’s law: W, the average time a customer spends at a Google property, whether that’s email, phone, calendar or web browser.

According to Comscore, Google received about 13 billion search queries in March 2014. This translates to 433.3 million queries per day, 18 million per hour, 300 thousand per minute and only 5,000 per second. A quick comparison to Bing looks like this:


Number of search queries
Timeframe Microsoft Google
Per month 3,600,000,000.00 13,000,000,000.00
Per day 120,000,000.00 433,333,333.33
Per hour 5,000,000.00 18,055,555.56
Per minute 83,333.33 300,925.93
Per second 1,388.89 5,015.43
Per millisecond 1.4 5


One wonders if Bing at any point exceeded Google’s 5,000 per second search rate. If yes, that’s good for Bing and bad for Google and it’s crucial to figure out why that jolt happened at that particular second. Investigating short bursts of higher-than-usual traffic leads to significant hints versus observing daily or monthly numbers.

Now consider Facebook. Facebook has both great arrival rate and time spent in “store.” But its customer arrival rate (λ) is not as high as Google’s. This is why all the top acquisitions and projects at Facebook target increasing the arrival rate. We visit Facebook a few times a day and stick around a little bit but then we quickly jump to a Google search.

Operation managers and entrepreneurs are more concerned with the throughput rate rather than the arrival rate. But the throughput rate is important only if there is arrival. Arrival is certainly a binary function without which there’s no usefulness. Once visitors arrive, the key metric to monitor is how fast they arrive, not how many.

Here are three implications of Little’s law as it applies to startups:

  1. For investors evaluating startups, it’s best to examine traffic figures at the lowest level of granularity possible. Even if the monthly uniques are low, surges in traffic at much smaller time intervals provide traces of higher value. The reverse is also true. Dips in arrival rates may suggest potential problems.
  2. For an entrepreneur, instead of focusing on the monthly stats, working on how to increase the searches per second is a healthier effort — particularly for those wanting to disrupt a certain market. The traffic numbers may be up and down and all over the place throughout the month, but it is the peaks of high traffic per second (or millisecond) that deserves the attention.
  3. It’s important to focus on why and how the influx of visitors surged in the smallest time frame available. Work to figure out ways to sustain that instead of focusing on monthly uniques.

Little’s law provides hints for social or viral growth, too, because in both cases, influence is spread out in short bursts as people visit the site/page/app almost all at the same time. Viral influx is the dream of a startup and after that, some level of stickiness is required to keep people around. But early traction trumps great content. Normalizing your metrics over time and looking at meaningful windows of time are a lot more useful than just looking at long-term averages.

If you’re hungry for analytical insights on traffic, traction and growth, look no further than queuing theory and particularly Little’s law. For those of you interested in the mathematical proof of Little’s law, here’s the link to Professor Little’s 2011 paper celebrating the 50th anniversary of his theory.


The Five Major Flows in Supply Chain

Supply Chain is the management of flows. There are Five major flows in any supply chain : product flow, financial flow, information flow, value flow & risk flow.

The product flow includes the movement of goods from a supplier to a customer, as well as any customer returns or service needs. The financial flow consists of credit terms, payment schedules, and consignment and title ownership arrangements. The information flow involves product fact sheet, transmitting orders, schedules, and updating the status of delivery.


Product Flow includes movement of goods from supplier to consumer (internal as well as external), as well as dealing with customer service needs such as input materials or consumables or services like housekeeping. Product flow also involves returns / rejections (Reverse Flow).

In a typical industry situation, there will a supplier, manufacturer, distributor, wholesaler, retailer and consumer. The consumer may even be an internal customer in the same organisation. For example in a fabrication shop many kinds of raw steel are fabricated into different building components in cutting, general machining, welding centres and then are assembled to order on a flatbed for shipment to a customer. Flow in such plant is from one process / assembly section to the other having relationship as a supplier and consumer (internal). Acquisition is taking place at each stage from the previous stage along the entire flow in the supply chain.

In the supply chain the goods and services generally flow downstream (forward) from the source or point of origin to consumer or point of consumption. There is also a backward (or upstream) flow of materials, mainly associated with product returns.



The financial and economic aspect of supply chain management (SCM) shall be considered from two perspectives. First, from the cost and investment perspective and second aspect based on from flow of funds. Costs and investments add on as moving forward in the supply chain.  The optimization of total supply chain cost, therefore, contributes directly (and often very   significantly) to   overall profitability.  Similarly, optimization of supply chain investment contributes to the optimisation of return on the capital employed in a company. In a supply chain, from the ultimate consumer of the product back down through the chain there will be flow of funds. Financial funds (Revenues) flow  from  the  final consumer, who  is usually the only source of “real” money in  a  supply  chain,  back  through   the other   links  in   the   chain   (typically retailers,  distributors,  processors  and suppliers).

In any organization, the supply chain has both Accounts Payable (A/P) and Accounts Receivable (A/R) activities and includes payment schedules, credit, and additional financial arrangements – and funds flow in opposite directions: receivables (funds inflow) and payables (funds outflow). The working capital cycle also provides a useful representation of financial flows in a supply chain. Great opportunities and challenges therefore lie ahead in managing financial flows in supply chains. The integrated management of this flow is a key SCM activity, and one which has a direct impact on the cash flow position and profitability of the company.


Supply chain management involves a great deal of diverse information–bills of materials, product data, descriptions and pricing, inventory levels, customer and order information, delivery scheduling, supplier and distributor information, delivery status, commercial documents, title of goods, current cash flow and financial information etc.–and it can require a lot of communication and coordination with suppliers, transportation vendors, subcontractors and other parties. Information flows in the supply chain are bidirectional. Faster and better information flow enhances Supply Chain effectiveness and Information Technology (IT) greatly transformed the performance.


A supply chain has a series of value creating processes spanning over entire chain in order to provide added value to the end consumer. At each stage there are physical flows relating to production, distribution; while at each stage, there is some addition of value to the products or services.  Even at retailer stage though the product doesn’t get transformed or altered, he is providing value added services like making the product available at convenient place in small lots.

These can be referred to as value chains because as the product moves from one point to another, it gains value. A value chain is a series of interconnected activities which are required to bring a product or service from conception, through the different phases of production (involving a combination of physical transformation and the input of various product services), delivery to final customers, and final disposal after use. That is supply chain is closely interwoven with value chain. Thus value chain and supply chain are complimenting and supplementing each other. In practice supply chain with value flow are more complex involving more than one chain and these channels can be more than one originating supply point and final point of consumption.

In chain at each such activity there are costs, revenues, and asset values are assigned. Either through controlling / regulating cost drivers better than before or better than competitors or by reconfiguring the value chain, sustainable competitive advantage is achieved.


Risks in supply chain are due to various uncertain elements broadly covered under demand, supply, price, lead time, etc.  Supply chain risk is a potential occurrence of an incident or failure to seize opportunities of supplying the customer in which its outcomes result in financial loss for the whole supply chain. Risks therefore can appear as any kind of disruptions, price volatility, and poor perceived quality of the product or service, process / internal quality failures, deficiency of physical infrastructure, natural disaster or any event damaging the reputation of the firm. Risk factors also include cash flow constraints, inventory financing and delayed cash payment. Risks can be external or internal and move either way with product or financial or information or value flow.

External risks can be driven by events either upstream or downstream in the supply chain:

  • Demand risks – related to unpredictable or misunderstood customer or end-customer demand.
  • Supply risks – related to any disturbances to the flow of product within your supply chain.
  • Environment risks – that originate from shocks outside the supply chain.
  • Business risks – related to factors such as suppliers’ financial or management stability.
  • Physical risks – related to the condition of a supplier’s physical facilities.

Internal risks are driven by events within company control:

  • Manufacturing risks – caused by disruptions of internal operations or processes.
  • Business risks – caused by changes in key personnel, management, reporting structures, or business processes.
  • Planning and control risks – caused by inadequate assessment and planning, and ineffective management.
  • Mitigation and contingency risks –  caused by not putting in place contingencies.



Supply chain management integrates key business processes from end user through original suppliers, manufacturer, trading, and third-party logistics partners in a supply chain. Integration is a critical success factor in a dynamic market environment and is prerequisite for enhancing value in the system and for effective performance of the supply chain by sharing and utilization of resources, assets, facilities, processes; sharing of information, knowledge, systems between different tiers in the chain and is vital for the success of each chain in improving lead-times, process execution efficiencies and costs, quality of the process, inventory costs, and information transfer in a supply chain. Integration leads to better collaboration for synchronized production scheduling, collaborative product development, collaborative demand and logistic planning. Also with increased information visibility and relevant operational knowledge and data exchange, integrated supply chain partners can be more responsive to volatile demand resulting from frequent changes in competition, technology, regulations etc. (capacity for flexibility). Integration is required not only for economic benefits but also for compliances in terms of social and community, diversity, environment, ethics, financial responsibility, human rights, safety, organizational policies, industry code of conduct, various national / international laws, regulations, standards and issues.


To achieve superior supply chain performance (cost, quality, flexibility and time performance) require multi-lateral integration :  Internal / External integration;  Functional integration, Geographical integration; Integration in Chains and networks; and Integration through IT. The integration even goes beyond to include supplier’s supplier and customer’s customer to leverage the power of the “network,” beyond their own.

Process Analysis 101 – Wait Times, Process Times, Little’s Law, Queuing Theory

Let’s take the example of a hospital. A patient checks in the hospital and then waits for some time to be examined by the nurses and doctors depending on the criticality of the patient. Or alternatively a patient might have requested to be checked only by a certain specialist doctor.

Below is a process flow diagram for the above mentioned

Patient Check-up process in Hospital

Now, calculate the overall wait and process times by doing a weighted average based on the number of patients arriving to the particular process flow. The calculations are as follows:

Avg Times - hospital.png

Lean Management says that all flow time should be value added time and there should be zero non-value added time (wait time). In this particular process, waiting time contributes to over 50% of the average flow time. So, we need to understand the reasons behind the high waiting time. To understand that, we will have to calculate what is the demand and capacity for each of the resources (doctors and nurses in this case).


The above demand and capacity is calculated through some given data about the hospital.


Waiting Time Causes.png

Theoretically, high waiting times are caused either because of high utilization  (less capacity (bottlenecks)) or because of high variation in the arrival or input flow. In turn, high variation can be caused because of huge changes in input volume or because of variation in time intervals of arrivals.



If capacity is the problem, then we add more capacity or do resource pooling. On the other hand, if high variation is the problem then we can take some design decisions such as appointment based checkups to schedule the arrival properly. In this particular case, we don’t have high variation in the input and hence we assume that the high waiting times are due to bottlenecks. Since doctors have are the least capacity resource, we say that doctors are the bottleneck.

Other inferences from the figure 2 are: the average flow  time (59 min) of patients going for a specialist examination is less than those going for a general checkup (68.4 mins). Ideally, patients who want to meet only a particular specialist doctor should have more flow time as they are putting adding stress to the system by requesting a specific resource. One can also check such design issues using this flow time, capacity and utilization calculations.

For a nice discourse on waiting times, please refer

Little’s Law: 

You can also Little’s law for other applications when arrival rates  and capacities or waiting times are given.

Step 1: Calculate the arrival rates of the process

Step 2: Calculate the waiting time and processing time at all the respective places in the process

Step 3: Calculate how much of the waiting time is congestion driven and how much is due to the variation in inter-arrival times and variation in arrival volumes

Little’s Law says: The mean number of jobs in the system = Arrival Rate * Mean Waiting Time In The System

It can also be said as:  the average queue length equals the average arrival rate times the average waiting time.

Alternatively: the average waiting time equals the average queue length times the average time between two arrivals.


So whenever you are standing in a customer line, Little’s law allows you to estimate how long you still have to wait. Estimate how many people are in front of you, estimate how long it takes for each customer to be served (more accurately, you would estimate the inverse of the departure rate), and multiply these. The graph represents how the total time spent increases as the arrival rate increases to approach the maximum server capacity (i.e. the maximum departure rate).

To understand the business applications of the Little’s law and queuing theory, refer and