Variance Study

1
Variance Study

• M. Peter Jurkat
• CS452/Mgt532 Simulation for Managerial Decisions
• The Robert O. Anderson Schools of Management
• University of New Mexico

2
Requirements

• Queuing theory says queues get long when
  utilization rises above .6-.7 based on average
  inter-arrival time and average service time
• No account of inter-arrival time variance
• To study effect of variance need to separate
  effects of inter-arrival mean from inter-arrival
  variance gt seek inter-arrival distribution for
  which variance can be changed w/o changing the
  mean

3
Examine Suitability of Distributions for Variance
Study
4
Strategy

• Queuing theory does not have formulas for
  performance measures (r, L, wq, ) in terms of
  inter-arrival variance gt need to simulate
• Banks4Example2-1SSQ.gp is the simples model
  change distribution in x-act GENERATE block to
  Gamma Distribution
• My version of GPSS World does not compile with a
  Gamma distribution gt use Excel to generate Gamma
  Distributions and enter into GPSS World as an
  empirical distribution

5
Reconcile Parameters

• Banks et al uses a different Gamma Distribution
  formulation that shows independence of variance
  from mean formulation in GPSS World and Excel
  does not
• GPSS World and Excel use the same formulation
  with different notation, but Excel does not show
  formulas for mean and variance
• Need to reconcile parameters in all three

6
Parameter Comparison
7
Excel Parameters in Terms of Distribution Mean
and Variance

• From argument of G() function aE aG bB
• From exponent of e we get bBqB 1/bE
• The mean, m, and variance, s2, are the same
  quantity regardless of the formulation
• From the Banks et al formulation m 1/qB and s2
  1/(bBqB2) 1/((bBqB) qB) 1/((1/bE)(1/m))
  bEm gt bE s2/m
• Also s2 1/(bBqB2) gt qB2 1/(bB s2) 1/m2 so
  bB s2 m2 and bB m2/ s2 aE
• These values of aE and bE in terms of m and s can
  now be used in Excel to develop Gamma
  Distributions whose means are independent of
  their variances see GammaDistributions.xls