Queueing Syst (2011) 68:353–360
DOI 10.1007/s11134-011-9240-3
Conditional inter-departure times from the M/G/s
queue
Casper Veeger · Yoav Kerner · Pascal Etman ·
Ivo Adan
Received: 9 May 2011 / Revised: 9 May 2011 / Published online: 1 July 2011
© Springer Science+Business Media, LLC 2011
Abstract We study the mean and the distribution of the time elapsing between two
consecutive departures from the stationary M/G/s queue given the number of customers left behind by the first departure is equal to n. It is conjectured that if the failure rate of the service time distribution is increasing (decreasing), then (i) the limit of
the mean conditional inter-departure time as n tends to infinity is less (greater) than
the mean service time divided by the number of servers s, and (ii) the conditional
inter-departure times are stochastically decreasing (increasing) in n for all n ≥ s.
Keywords Departure process · Failure rate
Mathematics Subject Classification (2000) 60K25
C. Veeger · P. Etman · I. Adan
Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513,
5600 MB Eindhoven, The Netherlands
C. Veeger
e-mail: [email protected]
P. Etman
e-mail: [email protected]
Y. Kerner ()
Department of Industrial Engineering and Management, Ben Gurion University of the Negev, Beer
Sheva, Israel
e-mail: [email protected]
I. Adan
Department of Quantitative Economics, University of Amsterdam, P.O. Box 19268, 1000 GG
Amsterdam, The Netherlands
e-mail: [email protected]
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Queueing Syst (2011) 68:353–360
Fig. 1 The sequence E[Dn ] for the M/G/5 with G = D (left) and G = C2 (right); the parameters of the
C2 distribution are set according to (4), ρ = 0.5 and μ−1 = 5 in all examples
1 Introduction
In this note we consider the standard M/G/s queue.
G(·)
∞ Let λ be the arrival rate and
λ
the service time distribution with mean μ−1 = 0 t dG(t), satisfying ρ ≡ μs
< 1.
The system is assumed to be in equilibrium. We are interested in the conditional
inter-departure times.
Inter-departure times are relevant, for example, in the context of queueing networks where the output of one queue provides the input to another queue; see [5, 9,
10]. Our interest in conditional inter-departure times arose out of work on aggregate
modeling of multi-processing work stations, where the conditional inter-departure
times are treated as work-in-process dependent process times in the aggregate model;
see [4, 7, 8].
Let the random variable Dn denote the time that elapses from a departure leaving
behind n customers in the system until the next departure. Since Dn is stochastically
smaller than the service time, its mean is bounded by μ−1 . We are interested in the
limit of E(Dn ) as n tends to infinity (if it exists) as well as distributional properties
of the sequence Dn . If the queue were infinitely long, customers would depart from
the system at rate μs. Hence, intuitively, one might expect that E[Dn ] tends to μ−1 /s
as n tends to infinity. Surprisingly, however, this does not appear to be the case,
and the first conjecture states that the limit (if it exists) is less (greater) than μ−1 /s
if the failure rate of the service time distribution is increasing (decreasing). This is
illustrated in Fig. 1, where the sequence E[Dn ] is shown for s = μ−1 = 5 and ρ = 0.5.
The mean conditional inter-departure times in Fig. 1 have been obtained by discreteevent simulation. The left graph shows E[Dn ] in case of deterministic service times.
The right graph shows E[Dn ] for Coxian service times with two phases and squared
coefficient of variation c2 = 0.5 and c2 = 4, respectively.
The second conjecture relates the failure rate properties of the service time distribution to stochastic monotonicity of the sequence Dn . This conjecture is motivated
by the intuition that, in case of a decreasing failure rate, longer queues indicate longer
Queueing Syst (2011) 68:353–360
355
Fig. 2 Phase diagram of the
Coxian distribution with two
phases
ages of the ongoing services, which in turn indicate longer residual service times and
hence, longer inter-departure times. For the Coxian distribution with two phases, the
failure rate is decreasing (increasing) if c2 > 1 (< 1). Hence, Fig. 1 suggests that if
the sequence Dn is stochastically monotone for n ≥ s, then it should be stochastically
increasing (decreasing) if the failure rate of the service time distribution is decreasing
(increasing).
In the next section we first determine the limit of E[Dn ] as n tends to infinity
for the special case of Coxian service times with two phases and show that, indeed,
this limit is less or greater than μ−1 /s depending on the failure rate of the service
time distribution. Section 3 presents the conjecture on the limit of E[Dn ] for general
service time distributions and Section 4 is devoted to the conjecture on the stochastic
monotonicity of the sequence Dn .
2 Coxian service times
We consider the special case of Coxian service times with two exponential phases,
labeled phase 1 and 2; see Fig. 2. The parameter of the ith exponential phase is
denoted by μi , i = 1, 2, and the probability of bypassing the second service phase by
1 − p.
The M/C2 /s system can be described by a Markov process on a strip, with states
(n, i) where n denotes the number of customers in the system and i the number of
servers in phase 2. Let πn,i denote the equilibrium probability of state (n, i) and rn
the long-run number of departures per time unit leaving behind n customers. Thus,
rn =
s
πn+1,i μ1 (1 − p)(s − i) + μ2 i .
(1)
i=0
Further, let vi be the expected time until the next departure if all servers are busy and
i of them are in the second service phase. Clearly, by conditioning on the first service
event (i.e., completion of phase 1 or 2 by one of the servers), vi satisfies the recursion
vs =
1
,
μ2 s
vi =
μ1 p(s − i)
1
+
vi+1 ,
μ1 (s − i) + μ2 i μ1 (s − i) + μ2 i
i = 0, 1, . . . , s − 1,
the solution of which is
vi =
s
j =i
j
−1
1
μ1 p(s − k)
,
μ1 (s − j ) + μ2 j
μ1 (s − k) + μ2 k
k=i
i = 0, 1, . . . , s.
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Queueing Syst (2011) 68:353–360
The mean conditional inter-departure time can then be expressed as
E[Dn ] =
s
1 πn+1,i μ1 (1 − p)(s − i)vi + μ2 ivi−1 ,
rn
(2)
i=0
where v−1 = 0 by convention. Now we are interested in the limit of E[Dn ] as n tends
to infinity, for which we need the limiting behavior of πn,i . In [2] it is shown that πn,i
can be expressed as a linear combination of s + 1 geometric terms, i.e.,
πn = (πn,0 , . . . , πn,s ) =
s
n
Cm y m x m
,
n = s, s + 1, . . . ,
m=0
where Cm is a scalar and ym is the row vector (ym,0 , ym,1 , . . . , ym,s ). For each m =
0, 1, . . . , s, the geometric factor xm is characterized as the unique root on the interval
(0, 1) of the equation
2m − s
=
s
2λ
s
(1 − x) + μ1 (1 − p)x 2 − (μ1 + μ2 )x
√
μ1 x(1 − (1 − p)x)
and the corresponding vector ym is a non-null solution (which exists) of the set of
homogeneous linear equations
ym,i xm λ + (s − i)μ1 + iμ2
2
= ym,i λ + ym,i−1 μ1 p(s − i + 1)xm + ym,i μ1 (1 − p)(s − i)xm
2
+ ym,i+1 μ2 (i + 1)xm
,
for all i = 0, 1, . . . , s, where by convention, ym,−1 = ym,s+1 = 0. In [2] it is proved
that the geometric factors are ordered as x0 > x1 > · · · xs . Hence, we immediately
have, as n tends to infinity,
πn,i = C0 y0,i x0n + o x0n .
(3)
Substitution of (3) into (1) and (2), and by letting n tend to infinity, we can conclude
that the limit of the sequence E[Dn ] exists and that it is equal to:
Proposition 1
lim E[Dn ] =
n→∞
s
1 y0,i μ1 (1 − p)(s − i)vi + μ2 ivi−1 ,
r∞
i=0
where
r∞ =
s
i=0
y0,i μ1 (1 − p)(s − i) + μ2 i .
Queueing Syst (2011) 68:353–360
357
Fig. 3 limn→∞ E[Dn ] as a function of ρ and squared coefficient of variation c2 ; μ−1 = s in all examples
From the analysis above, it readily follows that in case of C2 service times, the
sequence Dn converges in distribution as n tends to infinity. In fact, by employing the
matrix-geometric theory [6], convergence of E[Dn ] and convergence in distribution
of Dn can be shown to hold true for the broader class of phase-type distributions.
3 Conjecture on the limit
We can now use the result in Proposition 1 to exactly calculate the limit of the sequence E[Dn ] for Coxian service time distributions with two phases. The mean and
squared coefficient of variation of the C2 service time distribution matches with μ−1
and c2 when the parameters are set to
c2 − 1/2
μ2 μ1
−
1
. (4)
,
μ
=
4μ
−
μ
,
p
=
μ1 = 2μ 1 +
2
1
μ1 μ
c2 + 1
This parameter set for the C2 distribution has been used in Fig. 3 to calculate the
limit of E[Dn ] as a function of the server utilization ρ and the squared coefficient of
variation c2 ; in all examples we have set μ−1 = s.
The dashed lines in Fig. 3 indicate the “intuitive” limit of E[Dn ], i.e., μ−1 /s = 1.
Surprisingly, Fig. 3 clearly shows that for ρ < 1, the limit of E[Dn ] is less than 1
if c2 < 1.0, and it is greater than 1 if c2 > 1.0. The deviation of the limit from the
intuitive value decreases as ρ increases, and the deviation is greater for s = 10 than
for s = 2. In general, a sufficient condition for the coefficient of variation to be smaller
(greater) than 1 is the distribution to have an increasing (decreasing) failure rate. As
we show next, for the Coxian distribution with two phases, having a coefficient of
variation smaller (greater) than 1 is equivalent to having an increasing (decreasing)
failure rate.
Proposition 2 The failure rate of a Coxian distribution with two phases is monotone.
The failure rate is increasing if c2 < 1, decreasing if c2 > 1 and constant if c2 = 1.
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Proof In case μ1 = μ2 , the failure rate function is given by
r(t) = μ1
μ2 − (1 − p)μ1 − pμ2 e(μ1 −μ2 )t
μ2 − (1 − p)μ1 − pμ1 e(μ1 −μ2 )t
and hence its derivative has the same sign as p(μ2 − (1 − p)μ1 ) (and in particular,
it is independent of t). Hence, r(t) is monotone. It is easy to verify that the sign of
r (t) is the same as the sign of 1 − c2 . In case μ1 = μ2 ,
p
r(t) = μ1 1 −
1 + pμ1 t
and thus r(t) is increasing, and 1 − c2 = 2p 2 /(1 + p)2 > 0.
This leads us to believe that the following result might hold for the M/G/s, provided the limit of E[Dn ] as n tends to infinity exists:
Conjecture 1 In the M/G/s with mean service time μ−1 ,
lim E[Dn ] < (>)
n→∞
1
μs
if the failure rate of the service time distribution is increasing (decreasing).
4 Conjecture on the stochastic monotonicity
An increasing (decreasing) failure rate is equivalent to the residual lifetime being
stochastically decreasing (increasing) in its age. Hence, intuitively, while observing
a queueing system in which the service times have an increasing failure rate, a large
queue indicates long ages of the ongoing services, which in turns implies short residual service times. This is proved in [3] for the special case s = 1. More specifically,
it is shown in [3], and it can also be deduced from [1] that the conditional distribution of the age of the service time given the number of customers in the system is
a stochastically increasing sequence. The same intuition seems to apply in the multi
server setting at departure epochs, where a long queue indicates long ages of the other
s − 1 ongoing services. This leads us to the following conjecture.
Conjecture 2 In the M/G/s, if the failure rate of the service time distribution is increasing (decreasing), then the sequence Dn is stochastically decreasing (increasing)
in n ≥ s.
Well known examples of distributions with a monotone failure rate are the Erlang
distribution (with an increasing failure rate) and the Hyper-exponential distribution
(with a decreasing failure rate). Below we present some numerical examples, supporting the conjecture above. The numerical results have derived by using matrixgeometric theory [6].
Queueing Syst (2011) 68:353–360
359
Fig. 4 The sequence pn in the
M/E2 /2 and M/H2 /2 queues
Example 1 (The M/E2 /2 queue) Assume that s = 2 and that the service times follow
an E2 (μ) distribution, which has an increasing failure rate. For n ≥ 2, let pn be the
probability that at a departure epoch leaving behind n customers, the ongoing service
is in the first phase. We have
pn =
μπn+1,1
,
2μπn+1,2 + μπn+1,1
where πn,i is the steady state probability that there are n customers in the system
and i of the ongoing services are in the second phase. Let X be a random variable
having a C2 distribution with μ1 = μ2 = 2μ and p = 0.5. The distribution of X is
the distribution of the inter-departure time given that the ongoing service is in the
second phase. The inter-departure time Dn can be written as X + In Y , where X, Y
are independent, Y is exponential with rate 2μ and In is an indicator, independent
of X and Y , with P (In = 1) = 1 − P (In = 0) = pn . Clearly, the stochastic behavior
of the sequence Dn is determined by the behavior of the sequence pn . In particular,
Dn is stochastically decreasing if and only if the sequence pn is decreasing. The left
graph in Fig. 4 shows the sequence pn for various values of ρ. As we can see, the
sequence pn is decreasing.
Example 2 (The M/H2 /2 queue) Assume that s = 2 and that the service time
distribution is a mixture of two exponential distributions. This distribution has a
decreasing failure rate. The density of the service time distribution is given by
g(t) = αμ1 e−μ1 t + (1 − α)μ2 e−μ2 t , where without loss of generality, μ1 > μ2 . Let
pn be the probability that at a departure epoch leaving behind n customers, the ongoing service is in rate μ1 . We have
pn =
2μ1 πn+1,2 + μ2 πn+1,1
,
2μ1 πn+1,2 + (μ1 + μ2 )πn+1,1 + 2μ2 πn+1,0
where πn,i is the steady state probability that there are n customers in the system and i
customers are being served in rate μ1 . The distribution of the inter-departure time Dn
is a mixture of exponential distributions as well. Further, the probability weight given
to higher rates is increasing in pn and hence, Dn is stochastically increasing if and
only if pn is decreasing. The right graph in Fig. 4 shows the sequence pn for α = 0.5,
μ1 = 1 and μ2 = 0.2 and various values of ρ. As we can see, pn is decreasing.
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Queueing Syst (2011) 68:353–360
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