Approximations to the Erlang C queue length formula for M/M/n queues.
This program computes queue lengths for M/M/n queues with the Erlang C formula, and two approximations, one discovered by Sakasegawa (1977) and one documented, but not discovered, by Neil Gunther in Analyzing Computer System Performance With Perl::PDQ.
Here are the results:
servers util erlang gunther sakasegawa
1 0.000 0.000000 0.000000 0.000000
1 0.100 0.011111 0.011111 0.011111
1 0.200 0.050000 0.050000 0.050000
1 0.300 0.128571 0.128571 0.128571
1 0.400 0.266667 0.266667 0.266667
1 0.500 0.500000 0.500000 0.500000
1 0.600 0.900000 0.900000 0.900000
1 0.700 1.633333 1.633333 1.633333
1 0.800 3.200000 3.200000 3.200000
1 0.900 8.100000 8.100000 8.100000
1 0.990 98.010000 98.010000 98.010000
1 0.999 998.001000 998.001000 998.001000
2 0.000 0.000000 0.000000 0.000000
2 0.100 0.002020 0.002020 0.003947
2 0.200 0.016667 0.016667 0.024254
2 0.300 0.059341 0.059341 0.074837
2 0.400 0.152381 0.152381 0.176644
2 0.500 0.333333 0.333333 0.366151
2 0.600 0.675000 0.675000 0.715359
2 0.700 1.345098 1.345098 1.391387
2 0.800 2.844444 2.844444 2.894609
2 0.900 7.673684 7.673684 7.725338
2 0.990 97.517487 97.517487 97.568236
2 0.999 997.501750 997.501750 997.552285
4 0.000 0.000000 0.000000 0.000000
4 0.100 0.000088 0.000040 0.000765
4 0.200 0.002395 0.001282 0.007701
4 0.300 0.015878 0.009799 0.031726
4 0.400 0.060466 0.042036 0.091929
4 0.500 0.173913 0.133333 0.223403
4 0.600 0.430565 0.357353 0.497042
4 0.700 1.000193 0.884695 1.079036
4 0.800 2.385730 2.220054 2.468958
4 0.900 7.089779 6.868159 7.166418
4 0.990 96.812612 96.537437 96.871779
4 0.999 996.784391 996.503749 996.841140
8 0.000 0.000000 0.000000 0.000000
8 0.100 0.000000 0.000000 0.000064
8 0.200 0.000067 0.000004 0.001353
8 0.300 0.001516 0.000157 0.008640
8 0.400 0.012330 0.002099 0.034161
8 0.500 0.059044 0.015686 0.105650
8 0.600 0.209313 0.081999 0.286230
8 0.700 0.631407 0.342578 0.733982
8 0.800 1.830580 1.290202 1.940062
8 0.900 6.313797 5.441941 6.395396
8 0.990 95.812556 94.597231 95.825633
8 0.999 995.760755 994.509747 995.764233
16 0.000 0.000000 0.000000 0.000000
16 0.100 0.000000 0.000000 0.000002
16 0.200 0.000000 0.000000 0.000105
16 0.300 0.000019 0.000000 0.001277
16 0.400 0.000699 0.000003 0.007970
16 0.500 0.009019 0.000122 0.035135
16 0.600 0.062801 0.002709 0.127160
16 0.700 0.302577 0.037345 0.416536
16 0.800 1.219536 0.370723 1.361107
16 0.900 5.322091 3.275262 5.409913
16 0.990 94.403548 90.796343 94.308107
16 0.999 994.306571 990.529739 994.183115
32 0.000 0.000000 0.000000 0.000000
32 0.100 0.000000 0.000000 0.000000
32 0.200 0.000000 0.000000 0.000003
32 0.300 0.000000 0.000000 0.000081
32 0.400 0.000003 0.000000 0.000975
32 0.500 0.000291 0.000000 0.007169
32 0.600 0.007601 0.000002 0.039412
32 0.700 0.088712 0.000247 0.183844
32 0.800 0.642493 0.020298 0.815961
32 0.900 4.123770 1.024064 4.248781
32 0.990 92.431001 83.511763 92.159509
32 0.999 992.246601 982.601706 991.904841
64 0.000 0.000000 0.000000 0.000000
64 0.100 0.000000 0.000000 0.000000
64 0.200 0.000000 0.000000 0.000000
64 0.300 0.000000 0.000000 0.000002
64 0.400 0.000000 0.000000 0.000048
64 0.500 0.000000 0.000000 0.000739
64 0.600 0.000155 0.000000 0.007387
64 0.700 0.010296 0.000000 0.057112
64 0.800 0.224226 0.000032 0.392668
64 0.900 2.795970 0.067992 3.008045
64 0.990 89.688778 70.197190 89.173038
64 0.999 989.334082 966.873556 988.657359