Tables 5: Tests in banknotes' high places

Index

Table 5-1: D30F based on lognormal distribution
Table 5-2: Test A3 of 1,4C
Table 5-3: Test A3 results
Table 5-4: Test B3 of 1,9C
Table 5-5: Test B3 results
Table 5-6: D30F Distribution of 100,000 random numbers
Table 5-7: Test C3.4 of 1,2C
Table 5-8: Test C3.2 results
Table 5-9: Test C3.4 results
Table 5-10: Test C3.6 results
Table 5-11: Test C3 results
Table 5-12: Test D3 of 1,5C
Table 5-13: Test D3 results
Table 5-14: NBC[1]s of test D3
Table 5-15: Test E3 of 1,7C
Table 5-16: Test E3 results
Table 5-17: NBC[1]s of test E3
Table 5-18: Performances of 8 C2Ds in BHP
Table 5-19: Expected ratio of BLP to BHP
Table 5-20: Performances of 8 C2Ds in BP

Table 5-1

D30F (Decreasing 30 Frequencies) based on lognormal distribution

Num.	Calculation	Ration
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30	f(0.8)=0.4864 f(1.6)=0.2233 f(2.4)=0.1133 f(3.2)=0.0634 f(4.0)=0.0382 f(4.8)=0.0243 f(5.6)=0.0162 f(6.4)=0.0111 f(7.2)=0.0079 f(8.0)=0.0057 f(8.8)=0.0043 f(9.6)=0.0032 f(10.4)=0.0025 f(11.2)=0.0019 f(12.0)=0.0015 f(12.8)=0.0012 f(13.6)=0.0010 f(14.4)=0.0008 f(15.2)=0.0006 f(16.0)=0.0005 f(16.8)=0.0004 f(17.6)=0.0004 f(18.4)=0.0003 f(19.2)=0.0003 f(20.0)=0.0002 f(20.8)=0.0002 f(21.6)=0.0002 f(22.4)=0.0001 f(23.2)=0.0001 f(24.0)=0.0001	0.4818 0.2212 0.1122 0.0628 0.0378 0.0241 0.0160 0.0110 0.0078 0.0057 0.0042 0.0032 0.0024 0.0019 0.0015 0.0012 0.0010 0.0008 0.0006 0.0005 0.0004 0.0004 0.0003 0.0003 0.0002 0.0002 0.0002 0.0001 0.0001 0.0001
Total	1.0096	1.0000

¤ f(x)=1/x/√(2πe^(log(x))²). μ=0. σ=1.
¤ Ration of number 1 is 0.4846/1.0096=0.4818.
¤ f(x) of 24<x is enough small to be disregarded.
¤ This D30F is used on tests in BHP (Banknotes' High Places) where D[N] (Denomination of HN=N) is restocked with.

Table 5-2

Test A3 of 1,4C: The total QB (Quantity of Banknotes) based on minimum NBCs (the Number of Banknotes a payer Carries) to make Y1 to Y30 without change, using 1-yen banknotes and 4-yen banknotes

To make	NBC		D30F	QB
To make	[4]	[1]	D30F	QB
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12 ... Y29 Y30	0 0 0 1 1 1 1 2 2 2 2 3 ... 7 7	1 2 3 0 1 2 3 0 1 2 3 0 ... 1 2	0.4818 0.2212 0.1122 0.0628 0.0378 0.0241 0.0160 0.0110 0.0078 0.0057 0.0042 0.0032 ... 0.0001 0.0001	0.4818 0.4424 0.3366 0.0666 0.0779 0.0737 0.0650 0.0233 0.0243 0.0235 0.0215 0.0102 ... 0.0008 0.0009
Total			1.0000	1.7156

¤ 10-yen banknotes do not exist.
¤ QB is the sum of NBC[1] (NBC of 1-yen banknotes) and 1.06 times NBC[4], each of which is multiplied by D30F. e.g. QB to make Y6 is (2+1.06)*0.0241=0.0737.
¤ Total NBC[1] with D30F is 1*0.4818+2*0.2212+...+2*0.0001=1.4453. And total NBC[4] with D30F is 0*0.4818+...+7*0.0001=0.2550.

Table 5-3

Test A3 results: Total QBs and their Ex‰s, of 8 C2Ds

C2D	Total NBC		Total QB	Ex‰
C2D	[N]	[1]	Total QB	Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	0.8994 0.4459 0.2550 0.1590 0.1051 0.0725 0.0517 0.0378	0.6665 1.1276 1.4453 1.6703 1.8347 1.9578 2.0517 2.1251	1.6199 1.6003 1.7156 1.8388 1.9461 2.0347 2.1065 2.1652	12 0 72 149 216 271 316 353

¤ QB is the sum of NBC[1] and 1.06 times NBC[N].

Table 5-4

Test B3 of 1,9C: The total of minimum LCs to make Y1 to Y30 without change, using 1-yen banknotes and 9-yen banknotes

To make	NBM		D30F	LC
To make	[9]	[1]	D30F	LC
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12 ... Y29 Y30	0 0 0 0 0 0 0 0 1 1 1 1 ... 3 3	1 2 3 4 5 6 7 8 0 1 2 3 ... 2 3	0.4818 0.2212 0.1122 0.0628 0.0378 0.0241 0.0160 0.0110 0.0078 0.0057 0.0042 0.0032 ... 0.0001 0.0001	0.4818 0.4424 0.3366 0.2512 0.1890 0.1446 0.1120 0.0880 0.0117 0.0143 0.0147 0.0144 ... 0.0007 0.0008
Total			1.0000	2.1818

¤ 10-yen banknotes do not exist.
¤ LC is the sum of NBM[1] (the Number of 1-yen Banknotes Moved in cash transactions) and 1.5 times NBM[9], each of which is multiplied by D30F. e.g. LC to make Y11 is (2+1.5)*0.0042=0.0147.
¤ Total NBM[1] with D30F is 1*0.4818+2*0.2212+...+3*0.0001=2.1251. And total NBM[9] with D30F is 0*0.4818+...+3*0.0001=0.0378.

Table 5-5

Test B3 results: Total LCs and their Ex‰s, of 8 C2Ds

C2D	Total NBM		Total LC	Ex‰
C2D	[N]	[1]	Total LC	Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	0.8994 0.4459 0.2550 0.1590 0.1051 0.0725 0.0517 0.0378	0.6665 1.1276 1.4453 1.6703 1.8347 1.9578 2.0517 2.1251	2.0156 1.7965 1.8278 1.9088 1.9924 2.0666 2.1293 2.1818	122 0 17 63 109 150 185 215

¤ LC is the sum of NBM[1] and 1.5 times NBM[N].
¤ NBM[N]s and NBM[1]s are respectively the same as NBC[N]s and NBC[1]s of test A3 (table 5-3).

Table 5-6

D30F Distribution of 100,000 random numbers

Num.	Expectation	Distribution
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30	48,180 22,120 11,220 6,280 3,780 2,410 1,600 1,100 780 570 420 320 240 190 150 120 100 80 60 50 40 40 30 30 20 20 20 10 10 10	48,066 22,072 11,283 6,333 3,796 2,363 1,689 1,094 776 588 426 285 244 186 147 102 106 69 66 53 45 48 35 36 17 17 23 11 16 8
Total	100,000	100,000

¤ The average of the distributed numbers is 2.472.

Table 5-7

Test C3.4 of 1,2C: How the simulation of test C3.4 proceed in the case of 1,2C

NBC		NBR	To make	How
[2]	[1]	NBR	To make	How
...	...	...	...	...
0	1	-	Y1	1
0	0	3	Y2	2
2	0	-	Y1	2-1
1	1	6	Y10	2+2+2+2+2
2	1	...	...	...

¤ 10-yen banknotes do not exist.
¤ P (a Payer in cash transactions) tries every time to best reduce NBC.
¤ NBR (the Number of Banknotes Restocked with) is the least 2-yen banknotes to reach, the amount of shortage in the next transaction plus Y4. e.g. To make Y2 with no 2-yen and no 1-yen in the table, P restocks with three 2-yen banknotes. This NBR is (2-0+4)/2=3. And, to make Y10 with a 2-yen and a 1-yen, NBR is 6 that is the rounded up number of (10-3+4)/2=5.5.
¤ Other C2Ds also restock with D[N] similarly.
¤ On test C3.2, the plus amount is Y2. And on test C3.6, the margin is Y6.

Table 5-8

Test C3.2 results: Average QBs of 8 C2Ds

C2D	Avg. ABR	Max. NBC		Average NBC		Avg. QB
C2D	Avg. ABR	[N]	[1]	[N]	[1]	Avg. QB
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	Y4.143 Y5.222 Y5.801 Y6.483 Y7.214 Y8.018 Y8.834 Y9.739	16 11 8 7 6 5 4 4	1 2 3 4 5 6 7 8	1.827 1.091 0.749 0.566 0.454 0.379 0.325 0.285	0.500 0.996 1.505 1.997 2.501 3.001 3.509 3.988	2.437 2.153 2.299 2.597 2.982 3.403 3.853 4.290

¤ QB is the sum of NBC[1] and 1.06 times NBC[N].
¤ An average ABR (the Amount of Banknotes Restocked with) is inverse proportion to the frequency of restocking.

Table 5-9

Test C3.4 results: Average QBs of 8 C2Ds

C2D	Avg. ABR	Max. NBC		Average NBC		Avg. QB
C2D	Avg. ABR	[N]	[1]	[N]	[1]	Avg. QB
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	Y5.877 Y6.458 Y7.771 Y8.211 Y8.493 Y8.973 Y9.529 Y10.276	17 12 9 7 6 5 5 4	1 2 3 4 5 6 7 8	2.384 1.473 1.155 0.799 0.587 0.458 0.375 0.317	0.500 0.996 1.505 1.997 2.501 3.001 3.509 3.988	3.027 2.558 2.729 2.844 3.124 3.487 3.906 4.324

¤ QB is the sum of NBC[1] and 1.06 times NBC[N].
¤ Maximum NBC[1]s and average NBC[1]s are the same as test C3.2 (table 5-8).

Table 5-10

Test C3.6 results: Average QBs of 8 C2Ds

C2D	Avg. ABR	Max. NBC		Average NBC		Avg. QB
C2D	Avg. ABR	[N]	[1]	[N]	[1]	Avg. QB
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	Y7.721 Y8.680 Y8.807 Y10.071 Y11.728 Y11.347 Y11.150 Y11.441	18 12 9 8 6 6 5 4	1 2 3 4 5 6 7 8	2.924 1.915 1.290 1.051 0.931 0.673 0.504 0.399	0.500 0.996 1.505 1.997 2.501 3.001 3.509 3.988	3.600 3.026 2.872 3.111 3.488 3.715 4.044 4.411

¤ QB is the sum of NBC[1] and 1.06 times NBC[N].
¤ Maximum NBC[1]s and average NBC[1]s are the same as test C3.2 (table 5-8).

Table 5-11

Test C3 results: Average QBs and their Ex‰s, of 8 C2Ds

C2D	QB			Avg. QB	Ex‰
C2D	Test C3.2	Test C3.4	Test C3.6	Avg. QB	Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	2.437 2.153 2.299 2.597 2.982 3.403 3.853 4.290	3.027 2.558 2.729 2.844 3.124 3.487 3.906 4.324	3.600 3.026 2.872 3.111 3.488 3.715 4.044 4.411	3.021 2.579 2.633 2.851 3.198 3.535 3.934 4.342	171 0 21 105 240 371 526 684

¤ QBs come from tables 5-8, 5-9, and 5-10.
¤ Average QB of 1,2C is (2.437+3.027+3.600)/3=3.021.

Table 5-12

Test D3 of 1,5C: How the simulation of test D3 proceed in the case of 1,5C

NBC [1]	To make	How	LC
...	...	...	...
2	Y7	5+1+1 (5+5-1-1-1)	3.5 6
0	Y1	5-1-1-1-1	5.5
4	Y15	5+5+5	4.5
4	Y4	5-1 (1+1+1+1)	2.5 4
5	...	...	...

¤ 10-yen banknotes do not exist.
¤ P carries enough 5-yen banknotes to pay the amounts.
¤ P chooses every time the smallest LC according to NBC[1].
¤ LC is the sum of NBM[1] and 1.5 times NBM[5].

Table 5-13

Test D3 results: Average LCs and their Ex‰s, of 8 C2Ds

C2D	Average NBM		Avg. LC	Ex‰
C2D	[N]	[1]	Avg. LC	Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	1.236 0.824 0.618 0.494 0.412 0.353 0.309 0.275	0.667 1.134 1.396 1.749 1.934 2.207 2.350 2.570	2.522 2.370 2.324 2.490 2.552 2.737 2.813 2.982	85 20 0 72 98 178 211 283

¤ LC is the sum of NBM[1] and 1.5 times NBM[N].

Table 5-14

NBC[1]s of test D3

C2D	NBC[1]
C2D	Max.	Average
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	1 2 7 8 13 12 18 16	0.500 0.996 1.701 2.082 2.800 3.142 3.855 4.184

¤ Some NBC[1]s exceed 10 at times.

Table 5-15

Test E3 of 1,7C: How the simulation of test E3 proceed in the case of 1,7C

NBC [1]	To make	How	LCC [1]	LC
...	...	...	...	...
4	Y4	1+1+1+1 (7-1-1-1)	-	4 4.5
0	Y1	7-1-1-1-1-1-1	-	7.5
6	Y21	7+7+7	0	4.5
6	Y5	1+1+1+1+1 (7-1-1)	0 2	5 3.5
1	...	...	...	...

¤ 10-yen banknotes do not exist.
¤ P chooses every time the smallest LC according to NBC[1]. But if NBC[1] exceeds 5, P chooses the smallest LCC[1] (LC of 1-yen Change) before choosing the smallest LC. LCC[1] equals the number of 1-yen change.

Table 5-16

Test E3 results: Average LCs and their Ex‰s, of 8 C2Ds

C2D	Average NBM		Avg. LC	Ex‰
C2D	[N]	[1]	Avg. LC	Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	1.236 0.824 0.618 0.494 0.412 0.353 0.309 0.275	0.667 1.134 1.397 1.749 1.955 2.268 2.487 2.691	2.522 2.370 2.324 2.491 2.573 2.798 2.951 3.103	85 20 0 72 107 204 270 335

¤ Each of average LC is the same as or somewhat larger than LC of test D3 (table 5-13).

Table 5-17

NBC[1]s of test E3

C2D	NBC[1]
C2D	Max.	Average
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	1 2 6 6 7 7 8 8	0.500 0.996 1.700 2.081 2.724 3.044 3.551 3.988

¤ All the NBC[1]s are under good control.

Table 5-18

Performances of 8 C2Ds in BHP

C2D	Ex‰ of test				Total Ex‰
C2D	A3 0.05	B3 0.05	C3 0.45	E3 0.45	Total Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	12 0 72 149 216 271 316 353	122 0 17 63 109 150 185 215	171 0 21 105 240 371 526 684	85 20 0 72 107 204 270 335	122 9 14 90 173 280 383 487

¤ Ex‰ of tests A3, B3, C3, and E3 are mixed in the ratio of 5:5:45:45 to produce the total Ex‰. e.g. Total Ex‰ of 1,2C is 0.05*12+0.05*122+0.45*171+0.45*85=122.

Table 5-19

Expected ratio of BLP to BHP

Banknotes' Denomination	Currency	Re- stock	Expectation
Banknotes' Denomination	Currency	Re- stock	BLP	BHP
1, N	£	N	0	1*1
1, N	£	1	0	0
1, N, 10	A$, ¥, C$, MX$	10	1*4	0
1, N, 10	A$, ¥, C$, MX$	N	0	1*4
1, N, 10, 10N	W, L, €	10N	1*3	1*3
1, N, 10, 10N	W, L, €	10	1*3	0
1, N, 10, 10N, 100	Rp, Rs, CN¥, $, R$	100	2*5	0
1, N, 10, 10N, 100	Rp, Rs, CN¥, $, R$	10N	1*5	1*5
1, N, 10, 10N, 100, 100N	P	100N	2*1	1*1
1, N, 10, 10N, 100, 100N	P	100	2*1	0
Total			29	14

¤ Currencies and their banknotes' denominations are cited from measurements of banknotes.
¤ D[N₁] and D[N₂] of C3D currencies are not distinguished.
¤ The smallest D[1] of banknotes is described as "1" in the table.
¤ The smallest 0.1N denominations of some currencies are disregarded in classification.
¤ Restocked denominations are presumed, regardless of the reality, to be whether the largest or the second largest denomination, with their probabilities equal.
¤ Expectation of BLP (BHP) is the number of BLP (BHP) multiplied by the number of currencies. e.g. If a currency has 5 denominations from 1 to 100 like Rp (Indonesian Rupiah), and the largest 100 is restocked, 1's and 10's places become BLP. The expectation of BLP is 2 (the number of places) by 5 (the number of currencies), which is 10.

Table 5-20

Performances of 8 C2Ds in BP (Banknotes' Places)

C2D	Ex‰		Total Ex‰
C2D	BLP 29	BHP 14	Total Ex‰
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	184 18 2 118 99 35 147 360	122 9 14 90 173 280 383 487	164 15 6 109 123 114 224 401

¤ Ex‰ of BLP and BHP come from table 4-15 and table 5-18 respectively.
¤ Ex‰ of BLP and BHP are mixed in the ratio of 29:14 to produce the total Ex‰. e.g. Total Ex‰ of 1,2C is (184*29+122*14)/43=164.

1¤4

An Efficient Combination of the Denominations of a Currency
Tables 1: Measurements of coins
Tables 2: Tests in coins' places
Tables 3: Measurements of banknotes
Tables 4: Tests in banknotes' low places
Tables 5: Tests in banknotes' high places
Tables 6: Tests of C3Ds