Little's Law and the Coming Court Gridlock

April 21, 2020

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The COVID-19 pandemic has had far reaching impacts on virtually every sector of society and the court system is no exception. Across the country courts have either shut down completely or put an end to in-person court appearances to comply with social distancing guidelines. Eventually the courts will re-open and business will resume. But a shut down like the one we are experiencing could have devastating consequences on an already over-burdened court system. Even before COVID-19, courts had more cases than they could speedily handle, and their caseloads are increasing every year. The current crisis could lead to gridlock when courts do reopen. To explain why, we can look to Little’s Law.

Little’s Law is a theorem in queuing theory, a branch of probability theory that deals with lines. Little’s Law states that the average length of a line (L) is equal to the average throughput rate (λ) multiplied by the average time spent in the system (W). The formula is expressed as L=λW, but it can be adjusted algebraically to solve for any of the three variables. To find the average time in the system, the formula becomes W=L/λ.

Think of the line at the bank. To determine the average amount of time a customer will spend in the bank (W), all you need to know is how long the line is (L), and how quickly the tellers at the window are taking care of the customers (λ). If there are 10 customers in line (L) and the tellers can take care of 5 customers an hour (λ), then W=L/λ means the customer will spend an average of 10/5, or 2 hours in the bank.

Businesses often describe the variables a little differently to help estimate capacity. The (W) describes the lead time, or amount of time a piece of work takes from being ordered to being completed. The length of the line (L) becomes the amount of Work in Progress (WIP). And the throughput rate (λ) describes how many pieces of work can be completed in a given amount of time. W=WIP/λ means the amount of time it takes to complete a piece of work is equal to the amount of Work in Progress divided by the rate at which work is being completed.      

The application to the court system’s capacity is similar. How long a case takes from filing to resolution (W) is equal to the number of open cases (WIP) in the system, divided by the rate at which cases are being processed through the system. It is easy enough to see that because W=WIP/λ, if the number of cases (WIP) goes up, or the throughput rate (λ) goes down, the amount of time it takes to resolve a case (W) will increase. If there are 1000 open cases (WIP) and the courts can resolve 10 a day (λ), W=1000/10 means a case will average 100 days from start to finish. Now imagine there are 2000 open cases (WIP) and the courts can only resolve 5 a day (λ). W=2000/5 means a case will average 400 days from start to finish.

So how will Covid-19 likely impact those two variables? In the case of a bank, when it closes for the day it locks its doors, finishes with the customers inside and there is no problem. In effect WIP goes to zero. Courts are shutting or slowing down, but courts are not businesses. Existing cases are not leaving the court because their hearings have been stayed due to social distancing guidelines. And unlike a business, shutting the doors does not eliminate the line. Bank customers would just go home when they see the closed sign, not form a line outside the locked doors. Litigants, on the other hand, are not just leaving and coming back later. They are forming a line. In fact, COVID-19 is likely to create a dramatic increase in the number of new cases being filed. People are losing their jobs and filing employment cases, contracts are being violated and force majeure provisions being invoked, job losses are leading to late child support payments and enforcement actions, working from home may well increase the divorce and domestic violence rates, and a hundred other types of disputes are being caused by these unique and unprecedented circumstances. In short, old cases are staying around, new cases are likely to be filed faster than ever, and WIP is going up.

What about throughput rate (λ)? The news here is not any better due to throughput rate’s inverse relationship to throughput time. Throughput time is the average time it takes to complete one piece of work. To go back to the bank analogy, throughput time would be the average amount of time it takes a bank teller to help one customer, or in the case of the courts, the amount of time it takes the court to adjudicate one matter. Throughput rate, on the other hand, is the number of pieces of work that can be completed in a given amount of time. For the bank, this would be the number of customers the teller can serve in an hour, or the number of matters a court can adjudicate from start to finish in a year. Numbers can be useful to see the inverse relationship here. If a bank teller helps one customer every 10 minutes, throughput time is 10 minutes per customer. That means the bank teller can help 6 customers per hour, the throughput rate. If it takes the bank teller 20 minutes per customer, the teller’s rate is 3 customers per hour. The throughput time doubled, and the throughput rate was cut in half. It is an inverse relationship, as one goes up, the other goes down.

The compounding problem is that an increase in the case load (WIP) means an increase in the throughput time and therefore a decrease in the throughput rate (λ). In a bank there is a clearer distinction between WIP and throughput time. Remember that WIP is the number of customers in line and throughput time is the amount of time at the teller’s window. It makes intuitive sense that whether there are 2 people or 100 in line behind the customer at the window, the time it takes the teller to help the customer at the window is the same. Not so for the court, the number of open cases (WIP) directly impacts how much time it takes to adjudicate a case. Every case will require several court appearances, deliberation time from the judge, processing time for the clerks, etc. Because there are only so many hours in the day, the more cases there are the more days each case will need to be spread over to process it from beginning to end.

This is the utilization problem. As the percent of a system’s maximum resources being used increases, the throughput time increases on a logarithmic scale. In other words, the closer the courts are to maximum capacity the longer the throughput time and the slower the throughput rate. The number of open cases (WIP) going up makes the throughput rate (λ) go down, and the logarithmic nature of the relationship means that the closer the courts are to maximum capacity even small increases in the case load will have huge impacts on the throughput rate.

This creates a feedback loop that could quickly spiral the courts to gridlock. More open cases (WIP) means a lower throughput rate (λ). A lower throughput rate (λ) means fewer cases are removed from the court’s docket and the number of open cases (WIP) will grow even higher; which will lower the throughput rate (λ) further, and the closer the courts are to maximum capacity, the more impact any extra cases have on the throughput rate. Also remember that apart from λ growing WIP through this feedback loop, more cases are being added all the time, likely at an increased rate due to the unprecedented stresses of COVID-19.

Finally, back to Little’s Law: W=WIP/λ. The average time a case spends in the system (W) is determined by the number of open cases (WIP) divided by the throughput rate (λ). The number of open cases is increasing while the courts are shut down and COVID-19 is creating a host of new litigation. This will drive down the throughput rate, which will drive up the open cases faster and faster in a utilization feedback loop. The math is clear, when the numerator grows and a denominator shrinks, the quotient gets bigger. The quotient here is the amount of time it will take for the courts to resolve a case.

Attorneys can shorten the time frame and help their clients by using this calm before the storm wisely. Any part of a case timeline that can be completed now, should be completed now. When the courts do resume business, litigants will want to be as close to the front of the line as possible, because the line is growing longer, and it is moving slower. Little’s Law tells us it is only going to get worse.