MSCI 2150
August 11, 2018
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WAITING LINES and DECISION THEORY FORMULA SHEETS.
Formulas Sheet
1
Formulas for Basic Single Server (M|M|1) : (∞|∞|FCFS)
Assuming Poisson arrivals and Exponential service times.
λ = Average number of arrivals per time period.
µ = Average number served per time period.
c = Number of Servers.
1. System Utilization
U=
λ
µ
2. Expected Number of Busy Server = U
3. Expected Number of Idle Server = 1 - U
4. Prob of zero units in the system
5. Prob that a server is idle
P0 = 1 −
6. Prob of n unit in the system
7. L =
λ
µ
λ
µ
n
λ
Pn = P0
µ
λ
λ
=
µ
µ−λ
8. Lq =
λ2
µ(µ − λ)
9. W =
1
µ−λ
10. Wq =
2
Lq +
P0 = 1 −
λ
µ(µ − λ)
Formulas for Basic Multi Server (M|M|c) : (∞|∞|FCFS)
Assuming Poisson arrivals and Exponential service times.
1. System Utilization
U=
λ
cµ
1
2. P0 = n=c−1 n P 1 λ
1 λ c
cµ
+
µ
c! µ
cµ − λ
n=0 n!
c
λµ µλ
λ
3. L =
2 P0 + µ
(c − 1)! (cµ − λ)
MSCI 2150
August 11, 2018
L−
4. Lq =
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λ
µ
L
λ
Lq
6. Wq =
λ
7.
5. W =
Pn =

n
λ
1


P0 for n > c


n−c
 c!c
µ





1
8. Pw =
c!
3
1
n
n
λ
P0
µ
for n ≤ c
c λ
cµ
P0
µ
cµ − λ
Formulas for Undefined and Constant Service Times
.
1. P0 = 1 −
λ
µ
2
λ
+
µ
λ
2 1−
µ
λ2 σ 2
2. Lq =
3. L=Lq +
4. Wq =
λ
µ
Lq
λ
5. W=Wq +
6. U=
4
1
µ
λ
µ
Quadratic Equation
The solutions of the quadratic equation ax2 + bx + c = 0 correspond to the roots of the
function f (x) = ax2 + bx + c, since they are the values of x for which f (x) = 0. Here x
represents an unknown, while a, b, and c are constants with a not equal to 0. With the
above parameterization, the quadratic formula is:
√
−b ± b2 − 4ac
x =
2a
MSCI 2150
5
August 11, 2018
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Decision Theory
EM V (x) =
n
X
xi P (xi )
i=1
EOL(y) =
n
X
yi P (yi )
i=1
EV P I = EV W P I − Optimal EMV(under prior probabilities)
EV SI = EV W SI − Optimal EMV(under prior probabilities)
EV W SI = (Optimal EMV + ve Report) (Marginal Prob of + ve Report) +
(Optimal EMV - ve Report) (Marginal Prob of - ve Report)
EN GS = EV SI − Cost of Sampling
E =
EV SI
EV P I
H(Ai ) = α(row maximum) + (1 − α)(row minimum)
Hurwiczs Optimism Pessimism Criterion: The most well-known criterion is the Hurwicz
criterion, suggested by Leonid Hurwicz in 1951, which selects the minimum and the maximum payoff to each given action x. The Hurwicz criterion attempts to find a middle ground
between the extremes posed by the optimist and pessimist criteria. Instead of assuming
total optimism or pessimism, Hurwicz incorporates a measure of both by assigning a certain
percentage weight to optimism and the balance to pessimism.