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M/G/1-FIFO Queue with Uniform Service Times

An exact formula for the equilibrium M/U/1 waiting time density is now effectively known. What began as a numeric exploration became a symbolic banquet. Inverse Laplace transforms provided breadcrumbs in the trail; delay differential equations subsequently gave clear-cut precision. We also remark on tail probability asymptotics and on queue lengths.

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1 Case One ()

Set , and

A sequence of functions is defined iteratively as follows:

where the empty sum convention holds for and it is understood that for .  Since is a degree polynomial in with coefficients of the form , where , are rational numbers, the integrals , , all possess closed-form expressions.  Therefore the differential equation for can be solved exactly.  Stitching the fragments together gives rise to the density function

pictured in Figure 1, where denotes the greatest integer .

Let us illustrate in greater detail:

thus

implying

Continuing:

thus

implying

The pattern is maintained:

until (the equation length becomes fixed and replaces the rightmost ):

We count terms in each upon expansion, at least for .

When numerically evaluating large symbolic expressions, it is important to ensure that the working precision of floating point quantities is suitably high. To evaluate may require several hundred decimal digits because, for instance, the first two of the 348 numerator terms comprising are

upon setting .  The subtraction of nearly equal numbers, such as these, will lead to a horrific loss of precision unless appropriate care is taken.

2 Case Two ()

Set , and

A sequence of functions is defined iteratively as follows:

where is the Kronecker delta.  Stitching the fragments together gives rise to

pictured in Figure 2.

Let us illustrate in greater detail:

thus

implying

Beyond this point, the derivatives at match:

thus

implying

3 Case Three ()

Set and

A sequence of functions is defined iteratively as follows:

The role of

here is more pronounced than in Section 2: a jump discontinuity occurs in the density at

(as opposed to merely a sharp corner) .  Stitching the fragments together gives rise to

pictured in Figure 3.

We could do as before, indicating steps leading to and . A classical result due to Erlang [6, 7, 8, 9]:

renders this listing unnecessary (with , , ).  We wonder if such a formula (for the M/D/1 queue) possesses an analog for the M/U/1 queue.

4 Personal Notes

I taught a semester-long class in statistical programming at Harvard University (as a preceptor) for nearly ten years.  A favorite set of problems began with an M/M/1 example [10] involving a hospital emergency room (ER).

Patients (clients) arrive according to a Poisson process with rate .  One doctor (server) is available to treat them.  The doctor, when busy, treats patients with rate

.  More precisely: interarrival times are exponentially distributed with mean

and treatment lengths are exponentially distributed with mean

.  The ER is open 24 hours a day, 7 days a week.  Patients must wait until the doctor is free and are treated in the order via which they arrive.  Simulate the performance of the ER over many weeks.  What can be said about waiting time in queue per patient (excluding service time)?  Determine the mean, variance, mode and median of

to as high accuracy as possible.  Assume for this purpose that the expected number of arriving patients per hour is and that the expected number of treatment completions per hour (for a continuously busy ER) is .

Putting aside experiment in favor of pristine theory, the Laplace transform of service density is .  Formulas for the mean and variance of follow immediately from

consequently

giving and respectively.  An expression for the density of :

shows that the mode is ; integrating and solving the equation

implies

giving the median to be .

The aforementioned problem set continued with an M/G/1 example, involving the same ER parameters, but with Uniform[] treatment lengths.  Since was required, I arbitrarily chose  and .

Perform exactly the same simulation as before, except assume treatment lengths (in minutes) are uniformly distributed on the interval

.  Less is known about this scenario than the preceding (with exponential service times).

The thought of choosing or

did not occur to me until later.  I had imagined that numerical inversion of Laplace transforms was the only avenue available to reliably estimate the mode and median.

Just as

is the first moment of treatment lengths,

are the corresponding second and third moments.  Again, favoring pristine theory over experiment,

consequently [11, 12]

giving and respectively.  The mode (location of the density maximum, excluding ) occurs when

i.e.,

solving the equation

gives the median to be .

My classroom example deviates from the direction of research [13], which emphasizes heavy-tailed service time distributions.  Ramsay [14] discovered a remarkably compact formula – a single integral of a non-oscillating function over the real line – in connection with Pareto service times.  My formulas for uniform service times are sprawling by comparison.

The recursive solution of delay-differential equations is certainly not new [15] but application of such to queueing theory does not seem to be widespread.  Counterexamples include [16, 17]; surely there are more that I’ve missed.  Symbolics are mentioned in [13].

The waiting time probability for M/D/1 seems to decay exponentially (as do the other two cases).  The Cramér-Lundberg approximation [18] is applicable; alternatively, we have asymptotics [11, 19]

where is the unique root of .

Returning finally to M/U/1, let denote the number of patients in the system (both queue and service).  Define

Under equilibrium, with , and , , we have

consequently [20, 21]

variance

giving and respectively.  The final term for the variance is missing in [21].  The probability generating function seems to decay geometrically, but details surrounding the exact limit of successive ratios have not been verified [20].

I am grateful to innumerable software developers, as my “effective” formulas are too lengthy to be studied in any traditional sense.  Mathematica routines NDSolve for DDEs and InverseLaplaceTransform (for Mma version ) plus ILTCME [22] assisted in numerically confirming many results.  R steadfastly remains my favorite statistical programming language.  A student asked in 2006 for my help in writing a relevant R simulation, leading to the computational exercises described here and to my abiding interest in queues.

Figure 1: Waiting time density plot for Uniform[] service.
Figure 2: Waiting time density plot for Uniform[] service.
Figure 3: Waiting time density plot for Deterministic[] service.

References

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    Steven Finch
    MIT Sloan School of Management
    Cambridge, MA, USA
    steven_finch@harvard.edu