QUEUEING THEORY & how it can transform SC & Manufacturing performance

Queueing Theory is essentially about the calculation of average waiting times in front of capacity constrained work stations which, in the context of SCM, applies to material pieces waiting to be processed by a factory work centre, or shipped from a warehouse.

In general, the longer the average waiting time the longer is the queue of material pieces; this queue is usually known as stock or inventory.

The average waiting time in the queue is related to 4 factors:

1. The average processing time - each piece has to wait for the previous piece to be processed so the longer the processing time the longer the next piece has to wait before it can be processed.
2. The processing capacity - the greater the capacity the more the work centre can process in a given time period
3. The variability of material piece arrivals - if more material arrives in a period than there is processing capacity the wait time/queue/inventory will increase, if less arrives the inventory will diminish. If nothing arrives when there is no queue the capacity is lost.
4. The variability of processing capacity - around its average the rate of processing will vary, for example due to unplanned change-overs and processing breakdowns / slow-downs.

If the rate of material pieces arriving for processing is less than or exactly matches capacity, with no variability, there won't be a queue; if the rate of arrivals is greater than capacity the queue grows without limit.

Irrespective of the capacity utilisation, if there is any variability in the rate of arrivals or processing which leads, momentarily, to arrivals exceeding capacity (either because there is a surge in arrivals, maybe a batch, or because there is a short term loss of capacity) a queue will appear. Over time the queue will vary in length (including sometimes completely disappearing) depending on the interaction of the rates of processing v arrivals.

For any given level of utilisation, the average wait time in the queue grows directly in line with the aggregate variability of arrivals and processing.

But the degree of capacity utilisation has a non-linear impact upon the average wait time/queue length/inventory - as utilisation increases their growth accelerates until, at 100% utilisation they grow without limit because any capacity losses are permanent (unless additional capacity is made available).

These relationships are demonstrated below:

and the Kingman or VUT formula quantifies them as follows:

ave. wait time in Queue = Variability of arrivals & processing x Utilisation/1-utilisation x ave processing Time

Real factories, of course, are more complex than one work centre but the principles of QT can still be applied, tho' accurate formulae cannot.

How can queueing theory influence supply chain and factory management?

1. Use S&OP/IBP to plan capacities, especially of the bottle-neck, that exceed demand and never plan for 100% utilisation.
2. If an inventory reduction is required to, say, fulfil backordered demand, release additional capacity - quick but expensive
3. If you want to meet demand with less capacity and inventory you need to reduce variability, which can be achieved as follows:

• Reduce processing variability with TPM/TQM/Standard Work/5S/Mistake Proofing etc
• Reduce arrivals variability using split and smaller batches, the latter of which can be done, without losing capacity, by using quick change-overs/SMED
• Reduce both arrivals and processing variability by ceasing to interrupt master production schedules because these waste capacity, increase the lead-times of interrupted schedules and cause upstream inventory congestion.
• Stop using forecasts to drive master production schedules because the inevitable item level forecast inaccuracies (1) lead to backorder averting, but variability generating, MPS interruptions/expedites, irrespective of safety stock and time fences. These become permanently embedded, as do their negative effects (excess inventory, lead-time & capacity usage), not only due to continuing forecast inaccuracies but also because the resulting extended lead-times cause further service threats and yet more expediting.

If you're interested in learning how to implement a replenishment method that eliminates the need for variability generating schedule interruptions/expedites because of its insensitivity to weekly/monthly forecast inaccuracy, allows you to meet your planned service levels with less capacity and a c40% inventory reduction, is suitable for both stable and volatile demand patterns and short and long lead-times and doesn't obsolete your current ERP or require a proliferation of kan-bans, have a read of the following Lean isn't Lean without Pull, Carry On Expediting? Factory flow is nonlinear so don't use master production schedules

1. Even with world class forecast mix accuracy of 80%, some 80% of individual items will be found to be tracking with >40% error.