Description
This is a PPT describes losses incurred due to waiting.
Waiting lines- losses
Emergency room crowding and ambulance diversion
Macro economic trends driving this
• • • Increase in ER visits (14% from 1997 to 2000) Decrease in number of emergency departments (8.1% decline since 1994) Consequences: – Long wait times (see waiting time analysis) – Loss of throughput (requires new analysis)
20% of US hospitals are on diversion status for more than 2.4 hours per day
Analyzing loss systems
Demand Process
One trauma case comes in every 3 hours
(a=3 hours) a is the interarrival time Exponential interarrival times
Trauma center moves to diversion status once all servers are busy incoming patients are directed to other locations
Service Process
Patient stays in trauma bay for an average of 2 hours
(p=2 hours) p is the service time Exponential processing time
What is the probability that all trauma centers are busy?
Methodology
• • Define r = p / a Example: r= 2 hours/ 3 hours r=0.67 Recall m=3 Use Erlang Loss Table Find that P3 (0.67)=0.0255
• • •
•
Thus our trauma center will be on diversion status for 2.5 % of time
Erlang Function
Erlang Loss Table 1 0.0909 0.1667 0.2000 0.2308 0.2500 0.2857 0.3333 0.3750 0.4000 0.4118 0.4286 0.4444 0.4737 0.5000 0.5238 0.5455 0.5556 0.5652 0.5714 0.5833 0.6000 0.6154 0.6250 0.6296 0.6364 0.6429 0.6552 0.6667 0.6774 0.6875 0.6923 0.6970 0.7000 0.7059 0.7143 0.7222 0.7273 0.7297 0.7333 0.7368 0.7436 0.7500 0.7561 0.7619 0.7647 0.7674 0.7692 0.7727 0.7778 0.7826 0.7857 0.7872 0.7895 0.7917 0.7959 2 0.0045 0.0164 0.0244 0.0335 0.0400 0.0541 0.0769 0.1011 0.1176 0.1260 0.1385 0.1509 0.1757 0.2000 0.2237 0.2466 0.2577 0.2687 0.2759 0.2899 0.3103 0.3299 0.3425 0.3486 0.3577 0.3665 0.3836 0.4000 0.4156 0.4306 0.4378 0.4449 0.4495 0.4586 0.4717 0.4842 0.4923 0.4963 0.5021 0.5078 0.5188 0.5294 0.5396 0.5494 0.5541 0.5587 0.5618 0.5678 0.5765 0.5848 0.5902 0.5929 0.5968 0.6007 0.6082 3 0.0002 0.0011 0.0020 0.0033 0.0044 0.0072 0.0127 0.0198 0.0255 0.0286 0.0335 0.0387 0.0501 0.0625 0.0758 0.0898 0.0970 0.1043 0.1092 0.1192 0.1343 0.1496 0.1598 0.1650 0.1726 0.1803 0.1955 0.2105 0.2254 0.2400 0.2472 0.2543 0.2591 0.2684 0.2822 0.2956 0.3044 0.3087 0.3152 0.3215 0.3340 0.3462 0.3580 0.3695 0.3751 0.3807 0.3843 0.3915 0.4021 0.4124 0.4191 0.4224 0.4273 0.4321 0.4415 4 0.0000 0.0001 0.0001 0.0003 0.0004 0.0007 0.0016 0.0030 0.0042 0.0050 0.0062 0.0077 0.0111 0.0154 0.0204 0.0262 0.0294 0.0328 0.0351 0.0400 0.0480 0.0565 0.0624 0.0655 0.0702 0.0750 0.0850 0.0952 0.1058 0.1166 0.1221 0.1276 0.1313 0.1387 0.1499 0.1612 0.1687 0.1725 0.1781 0.1837 0.1949 0.2061 0.2172 0.2281 0.2336 0.2390 0.2426 0.2497 0.2603 0.2707 0.2775 0.2809 0.2860 0.2910 0.3009 m 5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0004 0.0006 0.0007 0.0009 0.0012 0.0020 0.0031 0.0045 0.0063 0.0073 0.0085 0.0093 0.0111 0.0142 0.0177 0.0204 0.0218 0.0240 0.0263 0.0313 0.0367 0.0425 0.0488 0.0521 0.0554 0.0577 0.0624 0.0697 0.0773 0.0825 0.0852 0.0892 0.0933 0.1016 0.1101 0.1187 0.1274 0.1318 0.1362 0.1392 0.1452 0.1541 0.1631 0.1691 0.1721 0.1766 0.1811 0.1901 6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0002 0.0003 0.0005 0.0008 0.0012 0.0015 0.0018 0.0021 0.0026 0.0035 0.0047 0.0056 0.0061 0.0069 0.0078 0.0098 0.0121 0.0147 0.0176 0.0192 0.0208 0.0220 0.0244 0.0282 0.0324 0.0354 0.0369 0.0393 0.0417 0.0468 0.0522 0.0578 0.0636 0.0666 0.0697 0.0718 0.0760 0.0825 0.0891 0.0937 0.0960 0.0994 0.1029 0.1100 7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0002 0.0003 0.0003 0.0004 0.0005 0.0008 0.0011 0.0013 0.0015 0.0017 0.0020 0.0027 0.0034 0.0044 0.0055 0.0061 0.0068 0.0073 0.0083 0.0100 0.0119 0.0133 0.0140 0.0152 0.0164 0.0190 0.0219 0.0249 0.0283 0.0300 0.0318 0.0331 0.0356 0.0396 0.0438 0.0468 0.0483 0.0506 0.0529 0.0577 8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0002 0.0003 0.0003 0.0004 0.0005 0.0006 0.0009 0.0011 0.0015 0.0017 0.0019 0.0021 0.0025 0.0031 0.0039 0.0044 0.0047 0.0052 0.0057 0.0068 0.0081 0.0096 0.0112 0.0120 0.0130 0.0136 0.0149 0.0170 0.0193 0.0210 0.0218 0.0232 0.0245 0.0274 9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0003 0.0004 0.0004 0.0005 0.0005 0.0007 0.0009 0.0011 0.0013 0.0014 0.0016 0.0018 0.0022 0.0027 0.0033 0.0040 0.0043 0.0047 0.0050 0.0056 0.0066 0.0077 0.0085 0.0089 0.0096 0.0102 0.0117 10 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0002 0.0003 0.0003 0.0004 0.0004 0.0005 0.0006 0.0008 0.0010 0.0013 0.0014 0.0016 0.0017 0.0019 0.0023 0.0028 0.0031 0.0033 0.0036 0.0039 0.0046 0.10 0.20 0.25 0.30 0.33 0.40 0.50 0.60 0.67 0.70 0.75 0.80 0.90 1.00 1.10 1.20 1.25 1.30 1.33 1.40 1.50 1.60 1.67 1.70 1.75 1.80 1.90 2.00 2.10 2.20 2.25 2.30 2.33 2.40 2.50 2.60 2.67 2.70 2.75 2.80 2.90 3.00 3.10 3.20 3.25 3.30 3.33 3.40 3.50 3.60 3.67 3.70 3.75 3.80 3.90
Probability{all m servers busy}=
r = p/a
rm m! Pm (r ) ? r1 r 2 rm 1? ? ? ... ? 1! 2! m!
• P3 (0.67)=0.0255 • On average, how many patients would be actually admitted to the trauma center per hour ?
• Patients admitted = patient arrival rate x • Probability that at least one unit is available
• = (1/3) x (1-0.025) • = (1/3) x 0.975 • = 0.325 patients per hour
• How many arriving patients are not admitted ?
• = (1/3) x 0.025 = 0.0083 patient per hour
doc_591455721.ppt
This is a PPT describes losses incurred due to waiting.
Waiting lines- losses
Emergency room crowding and ambulance diversion
Macro economic trends driving this
• • • Increase in ER visits (14% from 1997 to 2000) Decrease in number of emergency departments (8.1% decline since 1994) Consequences: – Long wait times (see waiting time analysis) – Loss of throughput (requires new analysis)
20% of US hospitals are on diversion status for more than 2.4 hours per day
Analyzing loss systems
Demand Process
One trauma case comes in every 3 hours
(a=3 hours) a is the interarrival time Exponential interarrival times
Trauma center moves to diversion status once all servers are busy incoming patients are directed to other locations
Service Process
Patient stays in trauma bay for an average of 2 hours
(p=2 hours) p is the service time Exponential processing time
What is the probability that all trauma centers are busy?
Methodology
• • Define r = p / a Example: r= 2 hours/ 3 hours r=0.67 Recall m=3 Use Erlang Loss Table Find that P3 (0.67)=0.0255
• • •
•
Thus our trauma center will be on diversion status for 2.5 % of time
Erlang Function
Erlang Loss Table 1 0.0909 0.1667 0.2000 0.2308 0.2500 0.2857 0.3333 0.3750 0.4000 0.4118 0.4286 0.4444 0.4737 0.5000 0.5238 0.5455 0.5556 0.5652 0.5714 0.5833 0.6000 0.6154 0.6250 0.6296 0.6364 0.6429 0.6552 0.6667 0.6774 0.6875 0.6923 0.6970 0.7000 0.7059 0.7143 0.7222 0.7273 0.7297 0.7333 0.7368 0.7436 0.7500 0.7561 0.7619 0.7647 0.7674 0.7692 0.7727 0.7778 0.7826 0.7857 0.7872 0.7895 0.7917 0.7959 2 0.0045 0.0164 0.0244 0.0335 0.0400 0.0541 0.0769 0.1011 0.1176 0.1260 0.1385 0.1509 0.1757 0.2000 0.2237 0.2466 0.2577 0.2687 0.2759 0.2899 0.3103 0.3299 0.3425 0.3486 0.3577 0.3665 0.3836 0.4000 0.4156 0.4306 0.4378 0.4449 0.4495 0.4586 0.4717 0.4842 0.4923 0.4963 0.5021 0.5078 0.5188 0.5294 0.5396 0.5494 0.5541 0.5587 0.5618 0.5678 0.5765 0.5848 0.5902 0.5929 0.5968 0.6007 0.6082 3 0.0002 0.0011 0.0020 0.0033 0.0044 0.0072 0.0127 0.0198 0.0255 0.0286 0.0335 0.0387 0.0501 0.0625 0.0758 0.0898 0.0970 0.1043 0.1092 0.1192 0.1343 0.1496 0.1598 0.1650 0.1726 0.1803 0.1955 0.2105 0.2254 0.2400 0.2472 0.2543 0.2591 0.2684 0.2822 0.2956 0.3044 0.3087 0.3152 0.3215 0.3340 0.3462 0.3580 0.3695 0.3751 0.3807 0.3843 0.3915 0.4021 0.4124 0.4191 0.4224 0.4273 0.4321 0.4415 4 0.0000 0.0001 0.0001 0.0003 0.0004 0.0007 0.0016 0.0030 0.0042 0.0050 0.0062 0.0077 0.0111 0.0154 0.0204 0.0262 0.0294 0.0328 0.0351 0.0400 0.0480 0.0565 0.0624 0.0655 0.0702 0.0750 0.0850 0.0952 0.1058 0.1166 0.1221 0.1276 0.1313 0.1387 0.1499 0.1612 0.1687 0.1725 0.1781 0.1837 0.1949 0.2061 0.2172 0.2281 0.2336 0.2390 0.2426 0.2497 0.2603 0.2707 0.2775 0.2809 0.2860 0.2910 0.3009 m 5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0004 0.0006 0.0007 0.0009 0.0012 0.0020 0.0031 0.0045 0.0063 0.0073 0.0085 0.0093 0.0111 0.0142 0.0177 0.0204 0.0218 0.0240 0.0263 0.0313 0.0367 0.0425 0.0488 0.0521 0.0554 0.0577 0.0624 0.0697 0.0773 0.0825 0.0852 0.0892 0.0933 0.1016 0.1101 0.1187 0.1274 0.1318 0.1362 0.1392 0.1452 0.1541 0.1631 0.1691 0.1721 0.1766 0.1811 0.1901 6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0002 0.0003 0.0005 0.0008 0.0012 0.0015 0.0018 0.0021 0.0026 0.0035 0.0047 0.0056 0.0061 0.0069 0.0078 0.0098 0.0121 0.0147 0.0176 0.0192 0.0208 0.0220 0.0244 0.0282 0.0324 0.0354 0.0369 0.0393 0.0417 0.0468 0.0522 0.0578 0.0636 0.0666 0.0697 0.0718 0.0760 0.0825 0.0891 0.0937 0.0960 0.0994 0.1029 0.1100 7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0002 0.0003 0.0003 0.0004 0.0005 0.0008 0.0011 0.0013 0.0015 0.0017 0.0020 0.0027 0.0034 0.0044 0.0055 0.0061 0.0068 0.0073 0.0083 0.0100 0.0119 0.0133 0.0140 0.0152 0.0164 0.0190 0.0219 0.0249 0.0283 0.0300 0.0318 0.0331 0.0356 0.0396 0.0438 0.0468 0.0483 0.0506 0.0529 0.0577 8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0002 0.0003 0.0003 0.0004 0.0005 0.0006 0.0009 0.0011 0.0015 0.0017 0.0019 0.0021 0.0025 0.0031 0.0039 0.0044 0.0047 0.0052 0.0057 0.0068 0.0081 0.0096 0.0112 0.0120 0.0130 0.0136 0.0149 0.0170 0.0193 0.0210 0.0218 0.0232 0.0245 0.0274 9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0003 0.0004 0.0004 0.0005 0.0005 0.0007 0.0009 0.0011 0.0013 0.0014 0.0016 0.0018 0.0022 0.0027 0.0033 0.0040 0.0043 0.0047 0.0050 0.0056 0.0066 0.0077 0.0085 0.0089 0.0096 0.0102 0.0117 10 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0002 0.0003 0.0003 0.0004 0.0004 0.0005 0.0006 0.0008 0.0010 0.0013 0.0014 0.0016 0.0017 0.0019 0.0023 0.0028 0.0031 0.0033 0.0036 0.0039 0.0046 0.10 0.20 0.25 0.30 0.33 0.40 0.50 0.60 0.67 0.70 0.75 0.80 0.90 1.00 1.10 1.20 1.25 1.30 1.33 1.40 1.50 1.60 1.67 1.70 1.75 1.80 1.90 2.00 2.10 2.20 2.25 2.30 2.33 2.40 2.50 2.60 2.67 2.70 2.75 2.80 2.90 3.00 3.10 3.20 3.25 3.30 3.33 3.40 3.50 3.60 3.67 3.70 3.75 3.80 3.90
Probability{all m servers busy}=
r = p/a
rm m! Pm (r ) ? r1 r 2 rm 1? ? ? ... ? 1! 2! m!
• P3 (0.67)=0.0255 • On average, how many patients would be actually admitted to the trauma center per hour ?
• Patients admitted = patient arrival rate x • Probability that at least one unit is available
• = (1/3) x (1-0.025) • = (1/3) x 0.975 • = 0.325 patients per hour
• How many arriving patients are not admitted ?
• = (1/3) x 0.025 = 0.0083 patient per hour
doc_591455721.ppt