CORRELATION IN OUTPUT ANALYSIS

10.2 CORRELATION IN OUTPUT ANALYSIS

To motivate the need for modeling correlations, we shall demonstrate that autocor- relation can have a major impact on performance measures, and so cannot always be ignored merely for the sake of simplifying models. A simple example will illustrate how autocorrelation is germane to queueing statistics (applications to manufacturing systems will be discussed later). The example compares two related models, such that autocorrelation is present in one, but not in the other.

Consider a simple workstation (similar to an M/M/1 queue) with job arrival rate l and processing (service) rate m (m > l), so that the system is stable with utilization m ¼ l=m < 1. It is known that in this simple system the steady-state mean sojourn time is E ½S M =M =1 Š ¼ 1=(m l ) (see Kleinrock [1975], chapter 3), and therefore the mean

198 Correlation Analysis waiting time in the buffer is E ½W M =M =1 Š ¼ 1=(m l ) 1 =m. Since all job arrivals

and processing times are mutually independent, all corresponding autocorrelations and cross-correlations are identically zero.

Next, modify the job arrival process to convert the workstation to a TES/M/1 type of system (see Melamed [1991, 1993], Jagerman and Melamed [1992a, 1992b, 1994]). That is, we replace the Poisson arrival process by a so-called (autocorrelated) TES process (see Section 10.3 for a description). The merit of TES processes is that they simultaneously admit arbitrary marginal distributions and a variety of autocorrelation functions. In particular, we can select TES interarrival processes with the same inter- arrival time distribution as in the Poisson process (i.e., exponential with rate l), but with autocorrelated interarrival times.

For a given TES process, let E ½W TES =M =1 Š denote the corresponding steady-state mean waiting time in the buffer. To gauge the impact of autocorrelations in the job arrival stream on mean waiting times, we observe the relative deviation

E ½W TES

=M =1 Š E ½W M =M =1 Š

d(r(1)) ¼

E ½W M =M =1 Š

as function of the lag-1 autocorrelation, r(1), of the TES job arrival process. These relative deviations provide a measure of the errors incurred by a modeler who models the TES/M/1 queue as an M/M/1 queue, thereby ignoring autocorrelation in the arrival process. Note that the TES arrival process above becomes a Poisson process when its autocorrelations vanish; in that case, d(r(1)) ¼ 0.

Table 10.1 displays the relative deviations of (10.2) for two representative cases: (1) m ¼ 1 and l ¼ 0:25 (light traffic regime with utilization u ¼ 0:25), and (2) m ¼ 1 and l ¼ 0:80 (heavy-traffic regime with utilization u ¼ 0:8). There is one exception: the column corresponding to r(1) ¼ 0 displays the exact mean waiting times for the baseline M/M/1 system (with respect to which the deviations, d(r(1)), are computed elsewhere in the table). Note that the relative deviation compares apples to apples, since distributions of both the service processes and the interarrival time processes (and hence the corresponding arrival rates) remain unchanged in both systems, and only the magnitude of r(1) is varied in the TES/M/1 system (it is, of course, always zero in the M/M/1 system).

The TES/M/1 mean waiting times were estimated by Monte Carlo simulation, since their analytic form is unknown. Note the dramatic nonlinear effect of the lag-1 auto- correlation on the relative deviation; in particular at r(1) ¼ 0:85, the relative deviation for the case of light traffic is about 9,000%, while for the case of heavy traffic it climbs to over 23,000%! Clearly, a naïve use of an M/M/1 model that ignores autocorrelations

Table 10.1 Relative deviations of mean waiting times for M/M/1 and TES/M/1 systems

Machine Lag-1 Autocorrelation of Job Interarrival Times, r(1) Utilization, u

Correlation Analysis 199 in its TES/M/1 counterpart produces unacceptable errors and is misleadingly over-

optimistic. For more information, see Livny et al. (1993). For additional studies of the performance impact of correlations in various random components of manufacturing systems, see Altiok and Melamed (2001).