Erlang Distribution
If interarrival or service times are not exponential, an Erlang random variable can often be used to model them. If T is an Erlang random variable with rate parameter R and shape parameter k, the density of T is given by
R(Rt)kϪ1eϪRt
f (t) ϭ ᎏᎏ
(k Ϫ 1)!
(t Ն 0)
and k E(T) ϭ ᎏᎏ
R
and
k var T ϭ ᎏᎏ
R2
Birth–Death Processes
For a birth-death process, the steady-state probability (pj) or fraction of the time that the process spends in state j can be found from the following flow balance equations:
( j ϭ 0)
( j ϭ 1)
( j ϭ 2) и и и ( jth equation)
p0l0 ϭ p1m1
(l1 ϩ m1)p1 ϭ l0p0 ϩ m2p2
(l2 ϩ m2)p2 ϭ l1p1 ϩ m3p3
(lj ϩ mj)pj ϭ ljϪ1pjϪ1 ϩ mjϩ1pjϩ1
The jth flow balance equation states that the expected number of transitions per unit time out of state j ϭ (expected number of transitions per unit time into state j). The solution to the balance equations is found from l0l1 и и и ljϪ1 pj ϭ p0 ᎏᎏ m1m2 и и и mj
( j ϭ 1, 2, . . .)
and the fact that p0 ϩ p1 ϩ и и и ϭ 1.
Notation for Characteristics of Queuing Systems pj L
Lq
Ls
W
Wq
Ws
l m ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ
steady-state probability that j customers are in system expected number of customers in system expected number of customers in line (queue) expected number of customers in service expected time a customer spends in system expected time a customer spends waiting in line expected time a customer spends in service average number of customers per unit time average number of service completions per unit time (service rate)
l r ϭ ᎏᎏ ϭ traffic intensity sm 1136
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The M/M/1/GD/∞/∞ Model
If r Ն 1, no steady state exists. For r Ͻ 1, pj ϭ r j (1 Ϫ r)
( j ϭ 0, 1, 2, . . .)
l
L ϭ ᎏᎏ mϪl l2
Lq ϭ ᎏᎏ m(m Ϫ l)
Ls ϭ r
1
W ϭ ᎏᎏ mϪl l
Wq ϭ ᎏᎏ m(m Ϫ l)
1
Ws ϭ ᎏᎏ m (The last three formulas were obtained from the L, Lq, and Ls formulas via the relation
L ϭ lW.)
The M/M/1/GD/c/∞ Model
If l
m,
1Ϫr
p0 ϭ ᎏᎏ
1 Ϫ rcϩ1 pj ϭ r jp0
( j ϭ 1, 2, . . . , c) pj ϭ 0
( j ϭ c ϩ 1, c ϩ 2, . . .) r[1 Ϫ (c ϩ 1)rc ϩ crcϩ1]
L ϭ ᎏᎏᎏ
(1 Ϫ rcϩ1)(1 Ϫ r)
If l ϭ m,
1
pj ϭ ᎏᎏ cϩ1 c
L ϭ ᎏᎏ
2
( j ϭ 0, 1, . . . , c)
For all values of l and m,
Ls ϭ 1 Ϫ p0
Lq ϭ L Ϫ Ls
L
W ϭ ᎏᎏ l(1 Ϫ pc)
Lq
Wq ϭ ᎏᎏ l(1 Ϫ pc)
1
Ws ϭ ᎏᎏ m Summary
1137
The M/M/s/GD/∞/∞ Model
For r Ն 1, no steady state exists. For r Ͻ 1,
1
ᎏ p0 ϭ ᎏᎏ s i iϭsϪ1
(sr)
(sr)
ᎏ
ᎏ
Α ᎏ! ϩ ᎏϪ r) i s!(1 iϭ0 (sr) j p0 pj ϭ ᎏᎏ j! (sr) j p0 pj ϭ ᎏᎏ s!s jϪs
(sr)s p0
P( j Ն s) ϭ ᎏᎏ s!(1 Ϫ r)
P( j Ն s)r
Lq ϭ ᎏᎏ
1Ϫr
P( j Ն s)
Wq ϭ ᎏᎏ sm Ϫ l l Ls ϭ ᎏᎏ m 1
Ws ϭ ᎏᎏ m l
L ϭ Lq ϩ ᎏᎏ m L
W ϭ ᎏᎏ l ( j ϭ 1, 2, . . . , s)
( j ϭ s, s ϩ 1, s ϩ 2, . . .)
(tabulated in Table 6)
The M/G/∞/GD/∞/∞ Model l L ϭ Ls ϭ ᎏᎏ m 1
W ϭ Ws ϭ ᎏᎏ m Wq ϭ Lq ϭ 0
The M/G/1/GD/∞/∞ Model s 2 ϭ variance of service time distribution l2s 2 ϩ r2
Lq ϭ ᎏᎏ
2(1 Ϫ r)
L ϭ Lq ϩ r
1
Ls ϭ l ᎏᎏ m Lq
Wq ϭ ᎏᎏ l 1138
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1
W ϭ Wq ϩ ᎏᎏ m 1
Ws ϭ ᎏᎏ m p0 ϭ 1 Ϫ r
Machine Repair (M/M/R/GD/K/K) Model l r ϭ ᎏᎏ m L ϭ expected number of broken machines
Lq ϭ expected number of machines waiting for service
W ϭ average time a machine spends broken
Wq ϭ average time a machine spends waiting for service pj ϭ steady-state probability that j machines are broken l ϭ rate at which machine breaks down m ϭ rate at which machine is repaired
Also,
pj ϭ
j r p
K
j r j!p
K
( j ϭ 0, 1, . . . , R)
j
0
j
0
ϭ
( j ϭ R ϩ 1, R ϩ 2, . . . , K)
R!R jϪR jϭK Lϭ
Α jpj
jϭ0
jϭK
Lq ϭ
Α ( j Ϫ R)pj
jϭR
jϭK
lϭ
ෆ
Α
jϭ0
jϭK
pjlj ϭ
Α l(K Ϫ j)pj ϭ l(K Ϫ L)
jϭ0
L
W ϭ ᎏᎏ l ෆ
Lq
Wq ϭ ᎏᎏ l ෆ
Exponential Queues in Series
If a steady state exists and if (1) interarrival times for a series queuing system are exponential with rate l; (2) service times for each stage i server are exponential; and (3) each stage has an infinite-capacity waiting room, then interarrival times for arrivals to each stage of the queuing system are exponential with rate l.
Summary
1139
The M/G/s/GD/s/∞ Model
A fraction ps of all customers are lost to the system, and ps depends only on the arrival rate l and on the mean ᎏ1ᎏ of the service time. Figure 21 can be used to find ps. m What to Do If Interarrival or Service Times
Are Not Exponential
A chi-square test may be used to determine if the actual data indicate that interarrival or service times are exponential. If interarrival and/or service times are not exponential, then
L, Lq, W, and Wq may be approximated by Allen–Cunneen formula.
For many queuing systems, there is no formula or table that can be used to compute the system’s operating characteristics. In this case, we must resort to simulation (see
Chapters 21 and 22).
Closed Queuing Network
Manufacturing and computer systems in which there is a constant number of jobs present may be modeled as closed queuing networks.
We let Pij be the probability that a job will go to server j after completing service at station i. Let P be the matrix whose (i Ϫ j)th entry is Pij .We assume that service times at server j follow an exponential distribution with parameter mj. The system has s servers, and at all times, exactly N jobs are present. We let ni be the number of jobs present at server i. Then the state of the system at any given time can be defined by an ndimensional vector n ϭ (n1, n2, . . . , ns). The set of possible states is given by SN ϭ {n such that all ni Ն 0 and n1 ϩ n2 ϩ и и и ϩ ns ϭ N}.
Let lj equal the arrival rate to server j. Since there are no external arrivals, we may set all rj ϭ 0 and obtain the values of the lj’s from the equation used in the open network situation. That is, iϭs lj ϭ
Α liPij
j ϭ 1, 2, . . . , s
iϭ1
Since jobs never leave the system, for each i, Αjϭs Pij ϭ 1. This fact causes the above jϭ1 equation to have no unique solution. Fortunately, it turns out that we can use any solution to help us get steady-state probabilities. If we define li ri ϭ ᎏᎏ mi then we determine, for any state n, its steady-state probability ⌸N(n) from the following equation: n rn1 rn2 и и и rn s
1
2
ᎏᎏ
⌸N(n) ϭ
G(N)
Here, G(N) ϭ ΑnʦSN rn1 rn2 и и и rns.
1
2 s Buzen’s algorithm gives us an efficient way to determine (in a spreadsheet) G(N). Once we have the steady-state probability distribution, we can easily determine other measures of effectiveness, such as expected queue length at each server and expected time a job
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