- Operational Models pdf.

But before we go on to talk about measuring we will show two simple applications of Little's little formula.

- Picture of arrival and departure generating system load diagram
- Area under system load is total response time: nR
- Average load, Q, times time interval, L is total response time: QL = nR
- n/L = X = throughput = \lambda = interarrival time.
- Therefore Q = XR = \lambda R

Little's formula holds only if the system is stable, which is defined by U < 0

- Infinite population of clients
- Is this actually possible? No, only in the limit.
- Only one possible distribution for interarrival time.
For a proof see this pdf.

- Demonstration: binomial -> Poisson -> exponential

- Finite population of clients - think-time model
- client works (called thinking), then server works, then client works (called thinking), etc.
- I would call this workload model a blocking model.

- To understand Figure 3.3 in the textbook. (p. 38)
- Solid curve real, dashed curve idealized

Top figure

- How throughput varies with load
- Linear region
- Saturation region
- Fall-off region: not in idealization

Bottom figure

- How response time varies with load
- Constant region
- Linear region
- Goes to infinity at the end of the linear region: system is becoming unstable, only hand-waving in idealization

- Finite population of clients in the think-time (blocking) model.

- X - Throughput
- R - average response time

- Interarrival time

As usual,

- Interval of time, L
- Number of arrivals, n

New

- Number of users, N
- Each user thinks before submitting the next request: average think time, Z.
- Number of users thinking, nt <= N; number of users waiting on systems, ns <= N. Because both are bounded the system is stable
- Average number of users in thinking, Nt. Average number of users waiting, Ns. Ns + Nt = N.
- NS is the load

Apply Little's formula

- Little's formula on the system: XR = Ns
- Little's formula on the users: XZ = Nt
- Therefore, X(R + Z) = N, which gives R = N/X - Z

Utilization

- Divide response time, ri, into queueing time, wi, and service time si: ri = wi + si.
- S is average serice time. S = (1/n) \sum si.
- S <= R
- The, by definition, utilization, U, is U = (1/L) \sum si = (nS) / L = XS = X \lambda.

Observations

Response Time

- R >= S, equality when U < 1.
- R = N/X - Z = (NS)/U - Z >= NS - Z, equality when U = 1

Throughput

- U = XS <= 1, which implies X = U/S <= 1/S.
- X(R + Z) = N, which implies X = N / (R + Z) >= N / (S + Z).

Now draw some pictures to see what's actually going on.

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