# Download PDF by Peter L Duren; Richard Askey; Uta C Merzbach; Harold M: A Century of mathematics in America

By Peter L Duren; Richard Askey; Uta C Merzbach; Harold M Edwards

ISBN-10: 0821801244

ISBN-13: 9780821801246

ISBN-10: 0821801309

ISBN-13: 9780821801307

ISBN-10: 0821801368

ISBN-13: 9780821801369

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**Extra info for A Century of mathematics in America**

**Example text**

I I I I l" I II I {i) (ii) P , . 1. Closed network of queues ~..! d, random variables; (iv) the sequences in (1), (il) and (iii) are mutually independent, In this network, there are two classes of Jobs: center 1 and class 2 Jobs at center 2. class i Jobs at Since each center sees only one class of Job, the process {Z(~):t~0} ca~ be defined by z(t) = (ql(t),q2(t)) where Ql(t) - number of Jobs waltln 8 or in service at center I at time t, and Q2(t) = number of Jobs waiting or in service at center 2 at time t.

2. Carry out the simulation of X for a fixed number, n, of cycles (having random length) defined by the successive returns to the state i0. 3. In each cycle measure all passage times of the marked Job. 2) n'~k_ 1 where 80"0 and 8k'Ul+.. '+~k" 4. 3) where n y'-n(f) ,, n -I ~ Yk (f) k'l and n %. k-1 5. Take as a i00(i-2y)% confidence interval (based on n cycles) for r(f) the interval 46 Here Zl_ym$-l(l-y), where @(') is the distribution function of a standardized and s n (mean zero, variance one) normal random variable, is the quantity where Sll = (=_l) -1 ~ (¥k(f)-~(£)) 2 , k=l and S22 = (n_l) -1 ~ (~k~) 2 k=l n s 2" -1 X • k"l Justification for this passage time simulation method appears in the next section.

EXAMPLES AND SIMULATION RESULTS To illustrate the marked Job method of the previous sections for estimation of passage times in closed networks of queues, we consider two examples. Descriptions of the networks and state vector definitions appear in this section, along with illustrative numerical results. 1. 1. Upon completion of ~ service at center i, in accordance with a binary-valued variable ~, the job Joins the tail of the queue in center 1 (when ~-i) or Joins the tail of the queue in center 2 for 8 service (when ~-0).

### A Century of mathematics in America by Peter L Duren; Richard Askey; Uta C Merzbach; Harold M Edwards

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