Recap: Maintaining <a target="_blank" rel="noopener noreferrer" href="https://www.vectorconsulting.in/en/research-and-publication/blogs/the-essence-of-managing-operations/">protective capacity (buffers) </a>is essential to ensure stable customer service levels in any operational environment characterised by variability. As utilisation rises, the available free capacity to address unexpected inflow diminishes, inevitably leading to the formation of queues.  While experienced operations managers often grasp this intuitively, they may need help to effectively communicate its importance to other departments, particularly those working in an environment focused on maximising utilisation to reduce production costs. The counterargument from the flag-bearers of cost reduction through higher utilisation is that higher utilisation reduces costs (using principles of cost absorption) and thereby increases profits. Another common rhetorical stance is that continuous emphasis on higher utilisation maintains the ‘pressure’ on workers, supposedly driving greater productivity. However, such perspectives often overlook the long-term consequences of excessive utilisation, including reduced service quality, customer dissatisfaction and eventual under-utilisation due to loss of customer orders. The challenge for operations managers lies in articulating the value of protective capacity in a scientifically grounded and compelling way. Without robust explanations, their advocacy for maintaining protective capacity risks being dismissed as a preference for ‘inefficiency’ despite its critical role in ensuring quality service and sustaining customer loyalty over time. This is where Kingman’s equation provides a valuable foundation. It scientifically explains why protective capacity is indispensable for achieving operational stability and organisational growth. By leveraging this principle, managers can advocate for a balanced approach that prioritises both utilisation and stability, ensuring sustainable development and long-term success.<h5> </h5><h5>Understanding Kingman’s equation</h5>Kingman’s equation is a foundational concept in queuing theory with a single workstation, which explores how cycle time (the time taken to process work through the single workstation) is affected by utilisation, protective capacity, environmental variability, and processing time. Kingman, a British mathematician, developed the following equation to illustrate this relationship:The formula is as follows.Cycle Time = (V)* (U/(1-U)) *(T)V is the measure of variability in both the arrival time of the job and the processing time or the touch time of a single job in the workstation.V is calculated as (Ca²+Ct²)/2.Ca is the coefficient of variability in arrival time, and Ct is the coefficient of variability in processing time.U is the measure of utilisation.1-U is the level of protective capacity available.T is the touch/processing time.<h5> </h5><h5>Key insights and intuition from Kingman's equation</h5><h6>Impact of utilisation on cycle time</h6> As utilisation increases, U approaches 1 (Utilisation~100%), the slack or protective capacity decreases, and waiting times rise sharply, indicating a more congested system. This means that for a given variability of an environment, the cycle time of an environment can go exponentially high when utilisation crosses a particular threshold. To illustrate in numbers, improving utilisation from 70% - 95% for any environment can lead to the following impact on the variable (U/1-U) of the equation. Let us calculate and seeU/(1-U) at 70% = 0.7/0.3 = 2.33U/(1-U) at 95% = 0.95/0.05 = 19 This means an attempt to increase utilisation by 25% can result in waiting time jumping by 800%!  Some job shop managers understand this impact more intuitively and manage their system using a monthly planning system. As the month progresses, the cycle starts ballooning towards mid-month. By the end of the month, they will try to throttle further material into the shop and focus on flushing out the excess WIP before allowing the intake in the next month. The graph below gives a visual imagery of the equation.  <figure class="image"><img style="aspect-ratio:2500/2500;" src="https://web-assets.vectorconsulting.in/storage/editor_uploads//figure-1-queue-length-vs-utilisation-1734676216_1776947857.jpg" width="2500" height="2500"></figure><h6>Impact in high variability environment</h6>Let’s evaluate the impact of variability on the waiting times – we have both variabilities factored into the equation - Ca is the coefficient of variability of arrival time (inflow), and Ct is the coefficient of variability in processing time (outflow). To check the phenomenon with some numbers, say the variability of an environment has gone up from Month 1 as below.Month 1 - {for Ca = 0.3, Ct =0.4} , the calculated value V = 0.125 One of the sources of variability can be changes in product mix with different processing times for the products of the mix. This leads to the coefficient of variability in processing time (outflow) increasing to 0.8. To worsen it further, let’s say one of the products needs an additional inspection step before the process. When the product mix is skewed towards this product, it will induce additional variability in arrival times, and the coefficient of variability of arrival time (inflow) goes up to 0.7. So, in month 2,Month 2 - {for Ca = 0.7, Ct = 0.8}, the calculated value V = 0.565 This implies that for the same level of utilisation, the waiting time can jump up to 450% with changes in the variabilities of inflow and outflow. This means that when the product mix changes, even if the planned loading is done at the same level of utilisation, the system can deteriorate drastically for end customers, leading to the loss of customers themselves. Such drastic increases in lead times can lead to much expediting, further increasing the variability and leading to more deterioration. Many textile plants in India face this problem when they change to a "better margin product" with lower volume per product and many variabilities in processing time in crucial work centres like dyeing, weaving and garment stitching. The top management is usually at a total loss for a system collapse, even when loading was done precisely as before. Such changes in product mix and consequent delivery issues can impact output for a period, leading to even drastic reductions in profits. Loss of customers can be an added problem, bringing a company to its knees!  Similarly, in new product development environments – with more loading, the level of multitasking upstream of the constraint work centre can be very high, leading to a very high level of uncertainty in the arrival rate to the constraint area. This can drop the system's output and increase the overall cycle time. In some manufacturing environments, high variability in arrival times goes up with high WIP, which can lead to cherry-picking in upstream work centres, leading to arrival time uncertainty to the bottleneck. This is more so in environments with assembly points and de-synch between matching parts. The flow of the parts can add to arrival time variability, further bloating up the cycle time. In environments where the work centre is essentially a human being – a designer, an artist, or a job worker performing non-standard jobs are inherently high-variable in processing times. In a system where these work centres feed to each other (steps in a design process or a job shop), the variability of both arrival and processing becomes very high. In such high variability environments like job shops, ETO, or NPD, the exponential impact of rising waiting times can kick in at a much lower utilisation level than in low-variability (deterministic arrival and processing time) environments. This implies that for a high variability environment, the protective capacity required for stable processing time is even higher than for a low variability environment.<h5> </h5><h5>Key insights</h5><h6>Controlling variability reduces the waiting time and improves the service levels:</h6>One can conclude from above that reducing the variability of an environment can drastically improve customer service! One can bring things back to normal if one knows what to focus on! In many production systems, we can improve upon the arrival time variabilities and ensure smooth flow by regulating the WIP daily (rather than monthly /quarterly) and having clear priority signals in the system. Further improvement in variability can be achieved by improving the daily availability of raw materials or reducing rework in work centres. In such environments, just by regulating the variabilities, one can prevent the creation of surprise bottlenecks. This, in turn, can not only drastically improve the cycle time but also improve the output.<h5> </h5><h5>A small protective capacity can have a dramatic positive impact</h5>In environments where bottleneck resources are skilled manpower and the system is severely congested with long cycle time, offloading a small portion of non-value-adding work from the skilled manpower can dramatically improve the cycle time and output of the system.<figure class="image"><img style="aspect-ratio:10407/2437;" src="https://web-assets.vectorconsulting.in/storage/editor_uploads//oops-1734676322_1776947938.png" width="10407" height="2437"></figure>In job shops, it makes sense not to discard old machines in constrained areas. They can add significant value in ensuring the stability of delivery lead times. In many skill-based operations, the attrition rates of the workers can create a destabilising effect on operations from time to time. Such plants need help to grow their business beyond a point. Such companies must investigate ways to arrest attrition or keep excess capacity to account for the attrition. Pharma labs in India usually face huge attrition rates, and plants need help with de-stabilised operations from time to time due to backlog built up in QC labs. Simple steps of releasing the capacity of analysts by removing non-value-adding work or providing help to low-skilled workers to offload routine non-analyst tasks can help maintain stability.<h5> </h5><h5>Actual usage & practical implementation</h5>While a valuable tool for analysing queueing dynamics, the Kingman equation is specifically designed for single work centres with variable arrival and processing rates. Even when we model a work centre with multiple resources and try to make the variabilities more deterministic, the effect of exponentially increasing waiting time dampens but still happens nonetheless.<figure class="image"><img style="aspect-ratio:2540/1984;" src="https://web-assets.vectorconsulting.in/storage/editor_uploads//oops-06-1734676357_1776948006.png" width="2540" height="1984"></figure><figure class="image"><img style="aspect-ratio:2540/1984;" src="https://web-assets.vectorconsulting.in/storage/editor_uploads//oops-07-1734676370_1776948095.png" width="2540" height="1984"></figure>Moreover, the equation currently does not adequately capture the complexities of natural manufacturing environments that often involve multiple interconnected work centres, dependent arrival rates, and varying conditions. Further, advanced simulations or analytical methods are required to accurately calculate critical inflexion points in these environments. These approaches must account for interdependencies and variability across the system. However, their effectiveness often hinges on the availability and accuracy of substantial operational data, which may be challenging to obtain in most environments. Hence, the primary value of the Kingman equations lies not in their direct application for pre-facto calculations but in building an intuitive understanding of an environment, particularly in a post-facto context. Thus, while these equations cannot directly model every environment, continuous daily data collection on protective capacity usage and cycle times can showcase the association between cycle time, variability, and utilization and help pinpoint inflection points essential for effective operational control.<h5> </h5><h5>Conclusion</h5>Kingman's equation is a powerful tool for helping operations managers articulate the trade-offs between utilisation and customer service in environments with inherent variability. By understanding and leveraging this equation, managers can advocate for protective capacity as a necessary buffer to maintain quality service, avoid excessive cycle times, and sustain long-term customer relationships. This knowledge empowers them to counter the short-sighted focus on high utilisation, showing that maintaining stability and flexibility often leads to greater long-term profitability and growth.

Key Takeaways

Learn how maintaining protective capacity ensures operational stability, reduces cycle times, and enhances customer satisfaction. Discover solutions to balance utilisation, variability, and profitability for sustainable growth.

Developing intuition of flow – Understanding variability, utilisation and cycle times

Recap: Maintaining <a target="_blank" rel="noopener noreferrer" href="https://www.vectorconsulting.in/en/research-and-publication/blogs/the-essence-of-managing-operations/">protective capacity (buffers) </a>is essential to ensure stable customer service levels in any operational environment characterised by variability. As utilisation rises, the available free capacity to address unexpected inflow diminishes, inevitably leading to the formation of queues.  While experienced operations managers often grasp this intuitively, they may need help to effectively communicate its importance to other departments, particularly those working in an environment focused on maximising utilisation to reduce production costs. The counterargument from the flag-bearers of cost reduction through higher utilisation is that higher utilisation reduces costs (using principles of cost absorption) and thereby increases profits. Another common rhetorical stance is that continuous emphasis on higher utilisation maintains the ‘pressure’ on workers, supposedly driving greater productivity. However, such perspectives often overlook the long-term consequences of excessive utilisation, including reduced service quality, customer dissatisfaction and eventual under-utilisation due to loss of customer orders. The challenge for operations managers lies in articulating the value of protective capacity in a scientifically grounded and compelling way. Without robust explanations, their advocacy for maintaining protective capacity risks being dismissed as a preference for ‘inefficiency’ despite its critical role in ensuring quality service and sustaining customer loyalty over time. This is where Kingman’s equation provides a valuable foundation. It scientifically explains why protective capacity is indispensable for achieving operational stability and organisational growth. By leveraging this principle, managers can advocate for a balanced approach that prioritises both utilisation and stability, ensuring sustainable development and long-term success.<h5> </h5><h5>Understanding Kingman’s equation</h5>Kingman’s equation is a foundational concept in queuing theory with a single workstation, which explores how cycle time (the time taken to process work through the single workstation) is affected by utilisation, protective capacity, environmental variability, and processing time. Kingman, a British mathematician, developed the following equation to illustrate this relationship:The formula is as follows.Cycle Time = (V)* (U/(1-U)) *(T)V is the measure of variability in both the arrival time of the job and the processing time or the touch time of a single job in the workstation.V is calculated as (Ca²+Ct²)/2.Ca is the coefficient of variability in arrival time, and Ct is the coefficient of variability in processing time.U is the measure of utilisation.1-U is the level of protective capacity available.T is the touch/processing time.<h5> </h5><h5>Key insights and intuition from Kingman's equation</h5><h6>Impact of utilisation on cycle time</h6> As utilisation increases, U approaches 1 (Utilisation~100%), the slack or protective capacity decreases, and waiting times rise sharply, indicating a more congested system. This means that for a given variability of an environment, the cycle time of an environment can go exponentially high when utilisation crosses a particular threshold. To illustrate in numbers, improving utilisation from 70% - 95% for any environment can lead to the following impact on the variable (U/1-U) of the equation. Let us calculate and seeU/(1-U) at 70% = 0.7/0.3 = 2.33U/(1-U) at 95% = 0.95/0.05 = 19 This means an attempt to increase utilisation by 25% can result in waiting time jumping by 800%!  Some job shop managers understand this impact more intuitively and manage their system using a monthly planning system. As the month progresses, the cycle starts ballooning towards mid-month. By the end of the month, they will try to throttle further material into the shop and focus on flushing out the excess WIP before allowing the intake in the next month. The graph below gives a visual imagery of the equation.  <figure class="image"><img style="aspect-ratio:2500/2500;" src="https://web-assets.vectorconsulting.in/storage/editor_uploads//figure-1-queue-length-vs-utilisation-1734676216_1776947832.jpg" width="2500" height="2500"></figure><h6>Impact in high variability environment</h6>Let’s evaluate the impact of variability on the waiting times – we have both variabilities factored into the equation - Ca is the coefficient of variability of arrival time (inflow), and Ct is the coefficient of variability in processing time (outflow). To check the phenomenon with some numbers, say the variability of an environment has gone up from Month 1 as below.Month 1 - {for Ca = 0.3, Ct =0.4} , the calculated value V = 0.125 One of the sources of variability can be changes in product mix with different processing times for the products of the mix. This leads to the coefficient of variability in processing time (outflow) increasing to 0.8. To worsen it further, let’s say one of the products needs an additional inspection step before the process. When the product mix is skewed towards this product, it will induce additional variability in arrival times, and the coefficient of variability of arrival time (inflow) goes up to 0.7. So, in month 2,Month 2 - {for Ca = 0.7, Ct = 0.8}, the calculated value V = 0.565 This implies that for the same level of utilisation, the waiting time can jump up to 450% with changes in the variabilities of inflow and outflow. This means that when the product mix changes, even if the planned loading is done at the same level of utilisation, the system can deteriorate drastically for end customers, leading to the loss of customers themselves. Such drastic increases in lead times can lead to much expediting, further increasing the variability and leading to more deterioration. Many textile plants in India face this problem when they change to a "better margin product" with lower volume per product and many variabilities in processing time in crucial work centres like dyeing, weaving and garment stitching. The top management is usually at a total loss for a system collapse, even when loading was done precisely as before. Such changes in product mix and consequent delivery issues can impact output for a period, leading to even drastic reductions in profits. Loss of customers can be an added problem, bringing a company to its knees!  Similarly, in new product development environments – with more loading, the level of multitasking upstream of the constraint work centre can be very high, leading to a very high level of uncertainty in the arrival rate to the constraint area. This can drop the system's output and increase the overall cycle time. In some <a target="_blank" rel="noopener noreferrer" href="https://www.vectorconsulting.in/en/solution-frameworks/custom-manufacturing/">manufacturing</a> environments, high variability in arrival times goes up with high WIP, which can lead to cherry-picking in upstream work centres, leading to arrival time uncertainty to the bottleneck. This is more so in environments with assembly points and de-synch between matching parts. The flow of the parts can add to arrival time variability, further bloating up the cycle time. In environments where the work centre is essentially a human being – a designer, an artist, or a job worker performing non-standard jobs are inherently high-variable in processing times. In a system where these work centres feed to each other (steps in a design process or a job shop), the variability of both arrival and processing becomes very high. In such high variability environments like job shops, <a target="_blank" rel="noopener noreferrer" href="https://www.vectorconsulting.in/en/industries/equipment-manufacturing/">ETO</a>, or <a target="_blank" rel="noopener noreferrer" href="https://www.vectorconsulting.in/en/solution-frameworks/research-and-development/">NPD</a>, the exponential impact of rising waiting times can kick in at a much lower utilisation level than in low-variability (deterministic arrival and processing time) environments. This implies that for a high variability environment, the protective capacity required for stable processing time is even higher than for a low variability environment.<h5> </h5><h5>Key insights</h5><h6>Controlling variability reduces the waiting time and improves the service levels:</h6>One can conclude from above that reducing the variability of an environment can drastically improve customer service! One can bring things back to normal if one knows what to focus on! In many production systems, we can improve upon the arrival time variabilities and ensure smooth flow by regulating the WIP daily (rather than monthly /quarterly) and having clear priority signals in the system. Further improvement in variability can be achieved by improving the daily availability of raw materials or reducing rework in work centres. In such environments, just by regulating the variabilities, one can prevent the creation of surprise bottlenecks. This, in turn, can not only drastically improve the cycle time but also improve the output.<h5> </h5><h5>A small protective capacity can have a dramatic positive impact</h5>In environments where bottleneck resources are skilled manpower and the system is severely congested with long cycle time, offloading a small portion of non-value-adding work from the skilled manpower can dramatically improve the cycle time and output of the system.<figure class="image"><img style="aspect-ratio:10407/2437;" src="https://web-assets.vectorconsulting.in/storage/editor_uploads//oops-1734676322_1776947919.png" width="10407" height="2437"></figure>In job shops, it makes sense not to discard old machines in constrained areas. They can add significant value in ensuring the stability of delivery lead times. In many skill-based operations, the attrition rates of the workers can create a destabilising effect on operations from time to time. Such plants need help to grow their business beyond a point. Such companies must investigate ways to arrest attrition or keep excess capacity to account for the attrition. Pharma labs in India usually face huge attrition rates, and plants need help with de-stabilised operations from time to time due to backlog built up in QC labs. Simple steps of releasing the capacity of analysts by removing non-value-adding work or providing help to low-skilled workers to offload routine non-analyst tasks can help maintain stability.<h5> </h5><h5>Actual usage & practical implementation</h5>While a valuable tool for analysing queueing dynamics, the Kingman equation is specifically designed for single work centres with variable arrival and processing rates. Even when we model a work centre with multiple resources and try to make the variabilities more deterministic, the effect of exponentially increasing waiting time dampens but still happens nonetheless.<figure class="image"><img style="aspect-ratio:2540/1984;" src="https://web-assets.vectorconsulting.in/storage/editor_uploads//oops-06-1734676357_1776947980.png" width="2540" height="1984"></figure><figure class="image"><img style="aspect-ratio:2540/1984;" src="https://web-assets.vectorconsulting.in/storage/editor_uploads//oops-07-1734676370_1776948078.png" width="2540" height="1984"></figure>Moreover, the equation currently does not adequately capture the complexities of natural <a target="_blank" rel="noopener noreferrer" href="https://www.vectorconsulting.in/en/solution-frameworks/custom-manufacturing/">manufacturing </a>environments that often involve multiple interconnected work centres, dependent arrival rates, and varying conditions. Further, advanced simulations or analytical methods are required to accurately calculate critical inflexion points in these environments. These approaches must account for interdependencies and variability across the system. However, their effectiveness often hinges on the availability and accuracy of substantial operational data, which may be challenging to obtain in most environments. Hence, the primary value of the Kingman equations lies not in their direct application for pre-facto calculations but in building an intuitive understanding of an environment, particularly in a post-facto context. Thus, while these equations cannot directly model every environment, continuous daily data collection on protective capacity usage and cycle times can showcase the association between cycle time, variability, and utilization and help pinpoint inflection points essential for effective operational control.<h5> </h5><h5>Conclusion</h5>Kingman's equation is a powerful tool for helping operations managers articulate the trade-offs between utilisation and customer service in environments with inherent variability. By understanding and leveraging this equation, managers can advocate for protective capacity as a necessary buffer to maintain quality service, avoid excessive cycle times, and sustain long-term customer relationships. This knowledge empowers them to counter the short-sighted focus on high utilisation, showing that maintaining stability and flexibility often leads to greater long-term profitability and growth.

Understand flow in manufacturing by exploring variability, utilisation, and cycle times. Insights from Vector Consulting Group to optimise operations. Read more here.