A thought about Queuing Theory

Queuing Theory is the statistical theory about the growth of lines -- what our British cousins tend to call queues. The theory can demand a fair amount of statistics, but if you have ever shopped in a supermarket you probably have some intuition about the way lines grow. If people get into the check out line in a market faster than they can be checked out,  the lines grow quickly and the time each customer waits in line also increases.

Lets give an example from health services. Consider a small hospital ward with 25 beds that has five patients admitted per day, each for a five day stay. On the day the ward opens, beds one through five are occupied; day 2 sees beds 6 through 10 occupied, and so on. On day 6, the five initial patients have been released,  and patients 26 through 30 are admitted into the first five beds. The process continues with no waiting.

Now consider that rather than five patients arriving per day, six arrive and all still need five days of care.

• Day 1: we see beds 1 through 6 occupied
• Day 2: beds 7 through 12 are occupied
• Day 3: beds 13 through 18 are occupied
• Day 4: beds 19 through 24 are occupied. 
• Day 5: bed 25 is occupied, 5 patients are told to wait until the next day to be admitted and at the end of the day 6 patients are discharged
• Day 6: beds 1 through 5 are occupied by the waiting patients, one new patient is given bed 6,  5 new patients are told to wait, and 6 patients are discharged at the end of the day
• Day 7: beds 7 through 11 are occupied by waiting patients, one new patient is given bed 12, 5 will to wait another day and 6 patients will be discharged
• Day 8: beds 13 through 17 are occupied by waiting patients, one new patient is given bed 18, 5 will have to wait another day and 6 patients discharged
• Day 9: beds 19 to 24 are occupied by waiting patients, all 6 new arrivals will have to wait another day,  and 6 patients discharged
• Day 10: bed 25 is occupied by one of the waiting patients. five waiting patients and all 6 new arrivals will have to wait, and six patients are discharged
• Day 11: beds 1 through 6 are occupied by waiting patients. five current waiting patients and all 6 arrivals will have to wait, and six patients are discharged

Thus after each five day cycle more patients are waiting for a bed to become free, and the average waiting time will increase with the number of cycles.

As long as the number of new patients arriving per day exceeds the number of available beds, the number of patients waiting for admission will increase and the average waiting time will lengthen.

There are two ways to deal with the increased demand. The better way is to provide more hospital beds. Less desirable is to make stays shorter, even if doing so patients risk more adverse health outcomes.

I think this is what has happened with the Veterans Administration. The wars in Iraq and Afghanistan have seen exceptional success in battlefield medicine, resulting in many lives being saved but many veterans returning home in need of medical attention from the VA. So too, there are increasing numbers of aging veterans of Vietnam, Korea and other military service. The VA could increase services accordingly (if appropriated the necessary budget increases), it could have reduced the resources given to meet the needs of each veteran seeking help,  or it could have made the waiting lines longer.

The Congress under the Constitution has the authority to appropriate funds for federal government operations. With the financial crisis in 2008, the Congress appropriated stimulus funding in 2008 and 2009. As the following graph shows, the Congress did not continue the stimulus and has reduced the rate of growth of government spending below that of the rate of growth of the economy.

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