Question

Dr Westin Dentist: What is the average number of patients waiting in queue? Dr. Weston is a dentist who oversees the training of new dental students in a local clinic serving the needs of the general public. The clinic has six dental chairs, each manned by a student. Patients arrive at the rate of 20 per hour according to a Poisson distribution. The average time required for a dental check-up is 15 minutes according to a negative exponential distribution. What is the average number of patients waiting in queue?

Answer 1

This is a M/M/s system with following parameters

$\lambda$ = 20 per hour

$\mu$ = 60/15 = 4 per hour

s = 6   (number of servers. here server is a dental chair manned by a student)

$\lambda$/$\mu$ = 20/4 = 5

Average number of patients waiting in queue, Lq = 2.938

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Method 1: Value of Lq can be directly read from the table of multi-channel queue system , corresponding to $\lambda$ /$\mu$ = 5 and M=6 , as highlighted in the following table

λμ Μ LG Po λμ Μ LG % λμ M LG Po 3.4 4.3 5.2 .003 ,005 005 4.4 3.5 5.3 .019 029 .032 .033 .033 .015 026 .029 .030 .030 .030 .011 .023 026 .027 .027 4.5 4 5 6 7 8 4 5 6 7 8 9 4 5 6 7 8 9 4 5 6 7 8 9 4 3.6 5.4 027 4.6 3.7 5.5 7 8 9 10 5 6 7 8 9 10 5 6 7 8 9 10 5 6 7 8 9 10 5 6 7 8 9 10 5 6 7 8 9 10 5 6 7 8 9 10 4.7 3.8 5 3.906 0.737 0.209 0.063 0.019 5.165 0.882 0.248 0.076 0.023 0.007 7.090 1.055 0.295 0.019 0.028 0.008 10.347 1.265 0.349 0.109 0.034 0.010 16.937 1.519 0.412 0.129 0.041 0.013 36.859 1.830 0.485 0.153 0.050 0.016 2.216 0.570 0.180 0.059 0.019 2.703 0.668 0.212 0.070 0.023 3.327 0.784 0.248 0.083 0.027 0.009 4.149 0.919 6 7 8 0.289 0.097 0.033 0.011 5.268 1.078 0.337 0.114 0.039 0.013 6.862 1.265 0.391 0.134 0.046 0.015 9.289 1.487 0.453 0.156 0.054 0.018 13.382 1.752 0.525 0.181 0.064 0.022 21.641 2.071 0.607 0.209 0.074 0.026 46.566 2.459 0.702 0.242 0.087 0.031 0011 2.938 0.810 0.279 0.101 0.036 0.013 3.536 0.936 0.321 0.117 0.042 0.015 .130 013 .014 .014 .006 2010 .012 .012 .012 .012 2005 .009 2010 .011 .011 .011 ,004 2008 2009 .010 .010 .010 2003 2007 2008 .009 .009 .009 2002 2006 .008 .008 2008 .008 .001 2005 2007 2007 .007 2007 007 .005 2006 .006 2007 .007 2007 2004 2005 .006 .006 2006 .006 6 7 8 9 10 11 6 7 8 9 10 11 12 6 7 8 9 10 11 12 6 7 8 9 10 11 12 6 7 8 9 10 11 12 6 7 8 9 10 11 12 6 7 8 9 10 11 12 6 7 8 9 10 4.301 1.081 0.368 0.135 0.049 0.018 5.303 1.249 0.422 0.155 0.057 0.021 0.007 6.661 1.444 0.483 0.178 0.066 0.024 0.009 8.590 1.674 0.553 0.204 0.077 0.028 0.010 11.519 1.944 0.631 0.233 0.088 0.063 0.012 16.446 2.264 0.721 0.266 0.102 0.008 0.014 26.373 2.648 0.823 0.303 0.116 0.044 0.017 56.300 3.113 0.909 0.345 0.133 5.6 .005 .005 .006 .003 .004 .005 .005 .005 .005 .005 .002 .004 ,004 .004 ,004 .005 .005 .002 .003 .004 .004 .004 .004 .004 .001 .003 .003 .004 .004 .004 .004 .001 .002 ,003 .003 .003 .003 .003 .001 .002 .003 .003 .003 .003 .003 .000 .002 .002 .003 .003 .008 .020 .023 .024 .025 .025 .005 .017 .021 .022 .022 .022 .002 .015 .019 .020 .020 .020 .013 .017 .018 .018 .018 .011 .015 .016 .016 .017 .009 .013 .014 .015 .015 .015 2008 .012 9 4.8 3.9 4.9 5.7 4.0 4 5 6 7 8 9 5 6 7 8 9 5 6 7 8 9 5 6 7 8 9 10 5 6 4.1 5.0 5.8 von 4.2 5.1 7 8 9 10 11 6 7 8 9 10 11 5.9 4.3

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Method 2: This is computed using Excel as shown below:

с D E F per hour per hour А B 1 A Mutliple-Server Waiting Line System 2 3 Input: 4. Arrival rate = 20 5 Service rate = 4 6 No. of servers, S = 6 7 8 Output: 9 PO = 0.0045 10 Rho = 0.833 11 Average number in the queue (Lq) = 2.938 12 13 14 15 16 17 Summation 1 n 1 2 3 4 5 6 7 6 18.5 39.33333333 65.375 91.41666667 113.1180556 00

EXCEL FORMULA:

C D E 1 per hour per hour A B A Mutliple-Server Waiting Line System 2 3 Input: 4 Arrival rate = 20 5 Service rate = 4 6 No. of servers, S = 6 7 8 Output: 9 PO = =1/((E16-E15)/(1-B4/(B5*B6)) +E15) 10 Rho = 1=B4/B5/B6 11 Average number in the queue (La) = 1=(E16-E15)*B9*B10/(1-B10)^2 12 13 14 15 16 17 18 19 n Summation 1 1 2 3 4 5 6 7 =($B$4/$B$5) D10/FACT(D10)+E10 =($B$4/$B$5) D11/FACT(D11)+E11 ($B$4/$B$5) D12/FACT(D12) +E12 ($B$4/$B$5) D13/FACT(D13)+E13 ($B$4/$B$5) D14/FACT(D14)+E 14 ($B$4/$B$5)-D15/FACT(D15)+E15