Calls to the 24-hour customer support line for Richman Financial Services occur randomly following a Poisson distribution with the following average rates during different hours of the day:

Time Period

Avg Calls Per Hour

Time Period

Avg Calls Per Hour

Midnight - 1 a.m.

2

Noon ? 1 p.m.

35

1 a.m. ? 2 a.m.

2

1 p.m. ? 2 p.m.

20

2 a.m. ? 3 a.m.

2

2 p.m. ? 3 p.m.

20

3 a.m. ? 4 a.m.

4

3 p.m. ? 4 p.m.

20

4 a.m. ? 5 a.m.

4

4 p.m. ? 5 p.m.

18

5 a.m. ? 6 a.m.

8

5 p.m. ? 6 p.m.

18

6 a.m. ? 7 a.m.

12

6 p.m. ? 7 p.m.

15

7 a.m. ? 8 a.m.

18

7 p.m. ? 8 p.m.

10

8 a.m. ? 9 a.m.

25

8 p.m. ? 9 p.m.

6

9 a.m. ? 10 a.m.

30

9 p.m. ? 10 p.m.

5

10 a.m. ? 11 a.m.

25

10 p.m. ? 11 p.m.

4

11 a.m. ? Noon

20

11 p.m. ? Midnight

2

The Richman's customer service representatives spend approximately 7 minutes on each call and are assigned to work 8-hour shifts that begin at the top of each hour. Richman wants to ensure that, on average, they can provide a 98% service level.

a. Determine the customer service schedule that allows Richman to achieve their service level objective using the fewest number of employees.

PART 2:

There is a planned upgrading programme for the national healthcare security in the Staffordshire County which is made up of the following districts:

East Staffordshire

Lichfield

Newcastle-under-Lyme

South Staffordshire

Stafford

Staffordshire Moorlands

Tamworth

There is a community hospital in each of the districts above. East Staffordshire district's community hospital will host the health record server for residents in the county. Dedicated underground cables will be laid to link up all the community hospitals in the county which will eventually be able to access the server hosted at East Staffordshire. Table 3-1 shows the costs incurred (in thousands of pounds) to connect the hospitals. Note that values in the lower triangle (grey cells) of Table 3-1 are omitted for brevity, and they are symmetric to the values in the upper triangle; for example, the cost between Newcastle-under-Lyme and East Staffordshire is the same as the cost between East Staffordshire and Newcastle-under-Lyme. Also note that it is not feasible for direct connection between some districts due to geographical constraints (e.g., between East Staffordshire and Lichfield

\fLet X and Y be independent random variables with cumulative distribution functions Fx and Fy, respectively. Show that (i) the cumulative distribution function of X +Y is Fxty (t ) = Fy(t - r)dFx (r); (ii) Fxty is continuous if one of Fx and Fy is continuous; (iii) X + Y has a Lebesgue density if one of X and Y has a Lebesgue density.Table 3-1 And Between East Lichfield Newcastle- South Stafford Staffordshire Tamworth Staffordshire under- Staffordshire Moorlands Lyme East 25 50 42 70 Staffordshire Lichfield 35 46 32 Newcastle-under- 40 35 Lyme South 60 Staffordshire Stafford 28 62 Staffordshire 36 Moorlands Tamworth