Question

Problem 18-4 The following information pertains to telephone calls to a motel switchboard on a typical Tuesday. Use Table 1 Service Rate Nunber of Operators 2 Incoming Rate (calls per minute) 1.6 (calls per minute per operator) 1.6 Period Morning Afternoon 2.7 9 1.5 3 Evening a. Determine the average time callers wait to have their calls answered for each period and the probability that a caller will have to wait for each period. (Round intermediate calculations and final answers to 3 decimal places.) Pw Period 0.33 Morning 0.509 Afternoon 0.702 Evening b. Determine the maximum line length for a probability of 91 percent. (Round your Intermediate calculations to 3 decimal places and round up your final answers to the next whole number.) Period Imax 0.888 Morning 5.025 Afternoon Evening 2.5

Answer 1

Calculations are done in the spreadsheet as follows:

A В C E F G Н K M N 1 Period л м Lq Wa Wq Pw specified % K Lmax P 2 Morning 0.625 1.6 1.6 2 1 0.333 0.2083 0.3333 0.500 0.91 0.5400 0.889 4. 3 Afternoon 1.111 0.2356 2.7 0.9 3 1.528 0.5659 0.5093 0.750 0.91 5.025 3.333 4 Evening 3.511 2.3407 0.7022 0.1538 1.5 0.6 3 2.5 0.833 0.91 10.268 5 Mathematical formulas: Lq value of Lq is obtained from Table 18.4 corresponding to /u and M 6 7 wa=1/(Μμ- ) 8 Wq Lq/A Pw- Wq/Wa 9 10 p=N(MH) K= (1-%) / (Lq*(1-p)) Lmax= In(K)/In(p) 11 12 13 14 15 16

TABLE 18.4 Infinite-source values for L and Pgiven /u and M P Nu M Po Po Νμ Νμ Μ 2.589 0.15 1 0.026 850 1.1 2 0.477 290 2.4 3 056 0.001 860 3 0.066 327 4 0.431 083 0.20 0.050 .800 4 0.011 332 5 0.105 .089 0.027 0,002 0,083 818 1.2 2 0.675 250 090 0.094 0.25 .750 3 294 7 0.007 091 0.004 778 4 0.016 300 2.5 3.511 045 0.003 074 0.30 1 0.129 .700 5 301 4 0.533 0.007 739 1.3 2 0.951 212 5 0.130 080 0.35 0.188 .650 3 0.130 264 6 0.034 082 271 0.009 2 0011 702 4 0.023 7 082 0.40 0.267 600 0.004 272 2.6 3 4.933 035 0.017 667 1.4 1.345 .176 4 0.658 065 0.368 0.177 0.45 1 550 236 5 0.161 072 0.043 0.024 .633 4 0.032 245 6 074 0.002 0.006 3 .637 5 246 7 0.011 074 0.50 1 0.500 .500 1.5 1929 .143 2.7 3 7.354 025 057 2. 0.033 .600 0.237 211 4 0.811 606 0.045 3 0.003 4 221 5 0.198 065 0,009 2.844 0.313 0.55 1 0.672 450 5 223 6 0.053 .067 0.045 569 1.6 .111 7 0.014 067 0.004 .187 576 3 2.8 3 12.273 .016 0.60 1 0.900 400 4 0.060 .199 4 1.000 050 0.059 538 5 0.012 201 5 0.241 058 060 3 0.006 548 1.7 4.426 .081 6 0.066 1.207 0.65 1 350 3 0.409 .166 7 0.018 061 0.077 .509 4 0.080 .180 2.9 3 27.193 008 1.234 3 0.008 521 5 0.017 .182 4 .044 7.674 0.70 1.633 300 1.8 053 5 0.293 052 0.098 2 481 3 0.532 146 6 0.081 054 3 0.011 .495 4 0.105 .162 7 0.023 055 0.75 .165 038 2.250 250 5 0.023 3.0 4 1.528 17.587 2 0.123 .455 1.9 2 .026 5 0.354 047 0.015 471 0.688 .128 6 0.099 049 0.80 3.200 200 4 0.136 .145 7 0.028 050 429 0.030 2 0.152 5 .149 8 0.008 050 0.019 .447 6 0.007 .149 3.1 4 1.902 032 0.85 4.817 150 2.0 3 0.889 .1111 5 0.427 042 0.187 0.174 2 404 4 130 6 0.120 044 0.040 0.035 0.024 425 5 .134 7 045 4 0.003 427 6 0.009 .135 8 0.010 .045 0.90 1 8.100 100 2.1 1.149 096 3.2 4 2.388 027 0.229 0.030 0.004 2. 379 4 0.220 .117 5 0.513 037 121 403 0.052 040 3 5 6 0.145 0043 4 406 6 0.012 .122 7 040 0.95 18.050 050 2.2 3 1.491 081 8 0.012 041 0.277 0.066 2 0.277 356 4 .105 3.3 4 3.027 023 0.037 383 5 .109 5 0.615 033 036 .037 386 .111 0.174 4 0.005 6 0.016 6 1.951 068 0052 1.0 2 0.333 333 2.3 3 7 0.045 364 4 0.346 .093 8 0.015 037 099 0.007 367 5 0.084 6 0.021 .100 (continued) LO co 0m LCK 02 st 02 LO 02 02s

Excel formulas:

Period

λ

μ

M

λ/μ

Lq

Wa

Wq

Pw

ρ

specified %

K

Lmax

Morning

1.6

1.6

2

=B2/C2

=1/3

=1/(D2*C2-B2)

=F2/B2

=H2/G2

=B2/D2/C2

0.91

=(1-K2)/F2/(1-J2)

=LN(L2)/LN(J2)

Afternoon

2.7

0.9

4

=B3/C3

1.528

=1/(D3*C3-B3)

=F3/B3

=H3/G3

=B3/D3/C3

0.91

=(1-K3)/F3/(1-J3)

=LN(L3)/LN(J3)

Evening

1.5

0.6

3

=B4/C4

3.511

=1/(D4*C4-B4)

=F4/B4

=H4/G4

=B4/D4/C4

0.91

=(1-K4)/F4/(1-J4)

=LN(L4)/LN(J4)

Note that in the above table, specified % has to be written as fraction. So, 91 percent, has to be written as 0.91

a)

Period	Pw
Morning	0.333
Afternoon	0.509
Evening	0.702

b)

Note that Lmax is rounded-up value to the next whole number

Period	Lmax
Morning	1
Afternoon	6
Evening	11