Networks of Queues: Myth and Reality
Guang-Liang Li and Victor O.K. Li
The University of Hong Kong
glli,vli@eee.hku.hk
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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Outline
1. “How Networks of Queues Came About”
2. Jackson Networks of Queues and Jackson’s
Theorem
3. Unsolved Mysteries
4. Counterexample 1: M/M/1 Queue with
Feedback
5. Counterexample 2: Two M/M/1 queues in
Tandem
6. Possible Behavior of Networks of Queues
7. Conclusion
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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1. “How Networks of Queues Came About”
• 2002, J. Jackson, “How networks of queues came
about,” Operations Research, vol. 50, no. 1, pp. 112113.
• 1957, J. Jackson, “Networks of waiting lines,”
Operations Research, vol. 5, no. 4, pp. 518-521.
• 1963, J. Jackson, “Jobshop-like queueing systems,”
Management Science, vol. 10, no. 1, pp. 131-142.
• After 1963, various generalizations and variations by
others.
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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2. Jackson Networks of Queues and
Jackson’s Theorem
• Jackson Network of Queues
• independent Poisson arrivals from outside
• independent exponential service times, also
independent of arrivals
• first-come-first-served
• once served at a queue, customer may either leave
network, or go to the same or another queue in the
network
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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 m*
m
m
 1*
mm
 M*
1
1
M
M
 km
MM
 11
 k*
k
k
 kk
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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Jackson’s Theorem
• Assumption: Network state (k1, k2, …, km) is a
stationary Markov process
• Theorem: In steady state, every queue in a
Jackson network behaves as if it was an M/M/m
queue in isolation, independent of all other queues
in the network.
Pk 1 , k 2 ,..., k M  Pk 1 Pk 2 ... Pk M
1
2
M
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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3. Unsolved Mysteries
• “product form solution”
• tandem network
– waiting times are dependent, cf., P.J. Burke, “The dependence of
delays in tandem queues,” Ann. Math. Statist., vol. 35, no. 2, June,
1964, pp. 874-875.
– but sojourn times are mutually independent, cf., E Reich, “Note on
queues in tandem,” Ann. Math. Statist., 34 338-341, 1963.
• M/M/1 with feedback behaves as if it was without feedback,
but
– with feedback: transition is impossible in small time interval if
feedback occurs
– without feedback: transition is always possible in any time interval
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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4. Counterexample 1: M/M/1 Queue with
Feedback
Diagnosing Jackson’s Proof
• m = 1, 2, …, M: labels of queues
• nm: number of servers at queue m
• m: service rate at queue m
• m: arrival rate of customers at queue m from outside network
• km: probability that customers go from queue m to queue k
•  m* = 1 - kkm: probability that customers leave network from
queue m
• i(k) = mink, ni, i = mink, 1
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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First Equation in Jackson’s Proof
Pk1, …, kM(t+h) = 1-(i)h – [i(ki)i]hPk1, …,
kM(t)
+i(ki+1)i  hPk1, …, ki+1, …, kM(t)
*
i
+iihPk1, …, ki-1, …, kM(t)
+j(kj+1)jijhPk1, …, kj+1, …, ki-1, …, kM(t)+o(h)
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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• M=1, n1=1, k1=k, 1=>0, 1=>0, 11=>0
• For k>1
Pk(t+h) = (1-h-h)Pk(t)+(1-)hPk+1(t)+hPk-1(t)
+o(h).
(1)
• X(t): number of customers waiting and being served
in the single-server queue at time t.
• Equation (1) is actually
P{X(t+h) = k} = P{X(t+h) = k|X(t) = k}P{X(t) = k}
+P{X(t+h) = kX(t) = k+1}P{X(t) = k+1}
+P{X(t+h) = k|X(t) = k-1}P{X(t) = k-1}+o(h) (2)
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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• Compare (1) with (2)
PX(t+h) = kX(t) = k = 1-h-h
• Replace k by k-1 in (1) and (2), and compare the obtained
equations.
PX(t+h) = k-1X(t) = k = (1-)h
• Replace k by k+1 in (1) and (2), and compare the obtained
equations.
P{X(t+h) = k+1|X(t) = k} = h
• Contradiction: the sum of the above probabilities is not equal
to one.
• X(t) is not a Markov process. Jackson’s theorem does not
hold.
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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Without Diagnosing Jackson’s Proof
• (2): time spent by X(t) in state k=2 until a transition to k=1.
• (2) is not exponential (to be shown) implies X(t) is not a
Markov process.
• actual service time of a customer: initial service time plus any
extra service time due to feedback
• (2) = d+p
• d>0: the (residual) actual service time of the departing
customer
• p>0: part of the actual service time of the other customer
• v(x): probability density function of p
• S: the (residual) exponential service time first expired in (2)
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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P{(2)<t =  0 P(2)<tp = x}v(x)dx
(3)
P{p = 0 = 1 implies v(x) = (x) (Dirac delta function), (2) = S,
(3) becomes


P(2)<t} = 0 P{S<t|p = x(x)dx
= P{S<tp = 0} = PS<t}
If 0<P{p = 0<1
P{(2)<t = P{(2)<t|p = 0P{p = 0
+  0+ P{(2)<tp = x}v(x)dx
= P{S<t}P{p = 0}+P{(2)<t|p>0}P{p>0}
For any 0<t<,
P{S<t}>P{(2)<t|p>0}
So (2) cannot be exponential.
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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5. Counterexample 2: Two M/M/1 queues in
Tandem
• All customers arrive at the first queue, go to the
second queue after service, and leave the network
form there.
• Jackson’s theorem in this case: corollary of Burke’s
theorem:
The output of the first queue is a Poisson Process at
the same rate as that of the arrival process.
The second queue is also an M/M/1 system.
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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Outline of Our Argument
1. The output of the first queue has both a marginal version, and
a non-marginal version (shall be demonstrated).
2. The non-marginal version is neither a Poisson process nor a
stationary process.
3. If the two queues are considered jointly as a network, the
arrival process at the second queue is the non-marginal, nonstationary version.
4. The second queue is not an M/M/1 queue and is unstable.
5. The state of this network is not stationary.
6. So Jackson’s theorem does not hold.
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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Output of M/M/1 Queue
Simulation (thought experiment) of stable M/M/1
queue in steady state
• Inter-departure time (t-s) is sampled in either case
below
• Case (a): server is busy at time s
- (t-s) is distributed as a service time
- color a line segment of length (t-s) red
- use “R” to represent the segment
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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• case (b): server is idle at time s
(t-s) is distributed as the sum of an idle time of the
server and a service time
color a segment of length (t-s) blue
use “B” to represent the segment
• sample path of the inter-departure time sequence
corresponds to a sequence of colored segments
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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Observation and Fact
• sequence of colored segments
RRRBRRRRBBBRRRBBBBBRR….
• segments of two colors: inter-departure times
follow two different distributions
• tendency for segments with the same color to
aggregate: Markov dependence
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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Non-Marginal Version of the Output
• the inter-departure time sequence: not i.i.d., not
stationary
• the corresponding departure process: not Poisson
process, not stationary
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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Marginal Version of the Output
• obtained by averaging out the impact of the state of the queue
• Experimental construction
divide interval (0, H) into N (H) consecutive, disjoint
subintervals of equal length
for all segments with length less than H, calculate the
frequencies that the lengths of the segments are in the small
intervals, regardless of their colors.
As H  , an exponential pdf with parameter equal to the
arrival rate is found
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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Marginal Version of the output (cont’d.)
• experimental construction (continued)
sample random variables independently,
regardless of the state of the queue, from the
constructed pdf
sampled random variables form an i.i.d.
exponential sequence
marginal version: the Poisson process
corresponding to the exponential sequence
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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What Does Burke’s Theorem Really Mean
Separate queues in tandem based on the marginal version so
as to treat them individually rather than jointly.
“It is intuitively clear that, in tandem queuing processes of the
type mentioned above, if the output distribution of each stage was
of such character that the queuing system formed by the second
stage was amenable to analysis, then the tandem queue could be
analyzed stage-by-stage insofar as the separate delay and queuelength distributions are concerned. Such a stage-by-stage
analysis can be expected to be considerably simpler than the
simultaneous analysis heretofore necessary. Fortunately, under
the conditions stated below, it is true that the output has the
required simplicity for treating each stage individually.”
P.J. Burke, “The output of a queueing system,” Operations
Research, vol. 4, pp. 699-714, 1956.
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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What if two queues in tandem are considered
jointly?
• the output of the first queue is the non-marginal,
non-stationary version
• the second queue is not M/M/1, and is not stable
• The state of the network is not a stationary process
• Jackson’s theorem does not hold
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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6. Possible Behavior of Networks of Queues
• Jackson network without loops:
not stationary if queues considered jointly
after separation based on the marginal version, queues
standing alone can be stable
• Jackson network with loops: not stationary
• network with renewal-type external arrivals and generally
distributed service times: not stationary in general
• tandem network with renewal-type external arrivals and
generally distributed service times: can be isolated and
isolated queues can be stable
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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7. Conclusion
• Jackson’s theorem does not hold, as shown by the
counterexamples.
• The assumption (i.e., network state is a stationary
Markov process) made by Jackson is invalid.
• All known “proofs” are based on this invalid
assumption.
• Jackson network is not stationary, unless queues can
be isolated.
• Generalizations and variations of Jackson networks
are questionable.
• Re-investigation of related issues is necessary.
 2003 G.L. Li and V. O.K. Li, The University of Hong Kong
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