analysis queue with autocorrelated times to failures (1)
TRANSCRIPT
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Analysis of Queues with Autocorrelated Times to
Failures
Bars Balcoglu
University of Toronto, Department of Mechanical and Industrial Engineering
5 Kings College Rd., Toronto, ON M5S 3G8, CANADA, [email protected]
David L. Jagerman
Rutgers University, RUTCOR
640 Bartholomew Rd., Piscataway, NJ 08854, USA [email protected]
Tayfur Altok
Rutgers University, Department of Industrial and Systems Engineering
96 Frelinghuysen Rd., Piscataway, NJ 08854, USA [email protected]
Abstract
In this paper, we study process completion time analysis and propose an accurate
approximation for the mean waiting time in queues with servers experiencing autocorre-
lated times to failure. To do this, we employ a three-parameter renewal approximation
that represents the autocorrelated times to failure stream. The analysis is exact in
the case of phase-type interruption processes if the arrival process is Poisson. We also
propose an accurate approximation for systems with renewal arrival processes if the
server interruption process is general.
Keywords and Phrases: Autocorrelation, M/PCT/1 Queues, M/G/1 Queues,
GI/PCT/1 Queues,GI/G/1 Queues, Waiting Time, Process Completion Time
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1 Introduction
The random phenomena observed in manufacturing systems arise primarily due to random
or semi-random processing times, and/or random machine failures/interruptions followed byrandom repair times. The randomness is the main cause of lack of productivity in manu-
facturing systems to which down time is a good contributor. While analyzing the impact
of machine failures on system performance, the general approach has been to assume a con-
stant processing time alongside random interruptions and repair times. In the literature,
these problems traditionally known as machine interference problemshave received consid-
erable attention (Dallery and Gerswhin, 1992), and queueing approach has been widely used
to analyze them. In this paper, we will study queueing systems in which servers encounter
autocorrelated times to failure. The autocorrelated failure process will be approximated
by a three-parameter renewal process (Balcoglu, Jagerman and Altok, 2005, Jagerman et
al., 2004) to construct an analytical model. We will asses the performance of the proposed
method by testing its accuracy in approximating the mean waiting time of the original
queueing system.
With machine interference problems, researchers typically try to optimize the size of the
repair-crew and try to come up with the optimal repair schedule. A recent paper by Iravani,
Duenyas and Olsen (2000) demonstrates the impact of unreliable machines on finished goods
inventory levels. Other works focus on mean response times, availability of the workstation,
average number of items in intermediate buffers, and average output rate (Altok, 1997,
Buzacott, 1972, Dogan-Sahiner and Altok, 1998, Nicola, 1986).
In this vast literature on machine interference problem, the autocorrelation that may
exist in interruption processes did not receive much attention. This is due to the fact that
introducing dependence in a process usually results in the loss of analytical tractability,
whereas the independence assumptions make resulting models easier to analyze.
Apart from dependent interarrival streams that arise frequently in high-speed integrated
telecommunication networks (Fendick, Saksena and Whitt, 1989), autocorrelated times to
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failure, which make the service times autocorrelated too, have been shown to have crucial
impact on system performances. For instance, Livny, Melamed and Tsiolis (1993) show that
positive lag-1 autocorrelation in the service process leads to increased mean waiting times.
More recently, Altok and Melamed (2001) simulate an M/G/1 workstation with determinis-
tic processing time and autocorrelated times to failure. They demonstrate that existence of
dependence in times to failure dramatically degrades the performance measures of interest,
such as flow time, customer service levels, and finished product levels. Consequently models
that ignore dependence (if any) in the underlying stochastic processes often become poor
representations of the corresponding real systems.
In this paper, we propose to approximate a positively autocorrelated interruption processby a three-parameter renewal process. These parameters summarize the important statis-
tics concerning the autocorrelation information, and help the approximating renewal process
represent the original process. This technique has been first proposed by Jagerman et al.
(2004) in approximating a single-source autocorrelated arrival process offered to worksta-
tions having general i.i.d. service time distributions. Its application has been extended by
Balcoglu, Jagerman and Altok (2005) to handle the superposition and splitting of autocor-
related arrival processes. Here, we will assume that the interarrival times of jobs arriving
at the workstation will be drawn from a general i.i.d. distribution. Hence, we will observe
to what extent the proposed approximation will capture the impact of dependence in the
interruption process on the mean waiting time.
The approximating renewal times to failure enables us to carry out the process completion
time analysis, first proposed by Gaver (1962) and Avi-Itzhak and Naor (1963). However,
in the literature, majority of the works on the process completion time analysis assume
Poisson failure processes, with possibly general i.i.d. processing and repair times, although
this assumption may not reflect the behavior of many real systems. This compromise at the
expense of having inaccurate predictions about the real system is made due to the fact that
the memoryless property of the exponential times to failure makes the analysis tractable,
whereas incorporating general i.i.d. times to failure imposes bigger challenge.
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Yet, Federgruen and Green (1986 and 1988) provide two prominent models with general
i.i.d. times to failure in single server queueing systems with Poisson arrivals. Federgruen
and Green (1986) derive approximations for the mean waiting time, probability of delay
and the steady-state distribution of the number in system for general i.i.d. times to fail-
ure. Federgruen and Green (1988), further, provide the exact solution for this system when
times to failure have phase-type distribution. In our analysis, since the proposed renewal
approximation yields a 2-state Hyper-exponential r.v., we could have used the results due
to Federgruen and Green (1988). However, we derive alternative solutions to compute these
probabilities, which we believe, are simpler and easier to implement. This technique can
be extended to cover cases for which the times to failure distribution can be modeled as
other forms of phase-type distributions as well. Our solution approach alongside that of
Federgruen and Green (1988) depends on the Poisson customer arrivals assumption, yet we
also extend our results with an approximate and accurate solution to study systems with
renewal customer arrival processes.
The rest of the paper is organized as follows. In Section 2, queueing systems experiencing
autocorrelated times to failure with Poisson arrivals are investigated and the corresponding
numerical examples are provided in Section 3. In Section 4, on the other hand, we propose
an approximate analysis of systems receiving renewal arrival processes.
2 Mean Waiting Time in M/PCT/1 Queues
In this section, we consider a single server queueing system receiving Poisson arrivals with
rate and fixed processing time x. The server encounters interruptions with autocorrelated
times to failure provided that it has jobs. It stays out of service throughout the repair
time that starts immediately after a failure. Repair times are assumed to have general i.i.d.
distributions with density function fD
(d). We assume that the server can fail only while it
is busy and resumes its operation from the point of interruption once a repair is completed.
Our objective is to compute the mean waiting time in this queue.
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We first construct an approximating renewal stream parameterized by X
, AE
, E
after
analyzing the autocorrelated times to failure data in line with Jagerman et al. (2004) and
Balcoglu, Jagerman and Altok (2005). Since we focus on bursty interruption processes
due to positive autocorrelation, AE turns out to be non-negative. This helps us express
the renewal times to failure r.v. as a 2-state Hyper-exponential (H2) r.v. with parameters
p, 1, 2, which is a special case of phase-type distributions. Note that H2 distributions are
extensively used to approximate distributions, whose squared-coefficient of variation exceeds
1. An H2 (p, 1, 2) r.v. is an exponential r.v with parameter1(2) with probabilityp(1p).Denoting its density function by g
R(t), the Laplace transform g
R(s) =
0 estg
R(t)dt of an
H2 r.v. is
gR
(s) = p 11+ s
+(1 p) 2
2+ s , (1)
and its parameters can be easily expressed in terms ofX
, AE
, and E
as
1 = X + E(1 + AE) +
(X+ E(1 + AE))
2
4XE2
,
2 = X + E(1 + AE) 1,
p =
X+ A
EE 2
1 2 . (2)
Note that so far we have approximated the autocorrelated failure process via a phase-
type interruption process. The remainder will be an exact analysis to obtain the performance
behavior of the underlying queueing system experiencing phase-type times to failure (machine
up times) denoted byU. Letting the times the machine is under repair (machine down times)
be denoted byD, the model will be an alternating stochastic process of the machines up and
down times, which stops when processing of a job is completed. Thus, we define the process
completion time, C(x), as the total time a job spends in processing, possibly augmented
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by down times due to failures. LetB(t, x) be its distribution function where x denotes the
constant processing time.
Under the process resumption after repair policy, a job being processed goes through a
sequence of up and down times. In this framework, the stopping rule is that the sum of the
up times is equal to the processing time, x. If K = k is the number of failures to occur
during the processing time of a job, then the process is said to be completed during the
k+ 1st up time. The first up time, U
1, is the leftover from the last up time of the previous
job, andU
k+1is the portion of the k +1st up time to be spent on the current job to complete
the process. In other words, U
1 and U
k+1 will respectively be the forward and backward
recurrence times if the system is operating in equilibrium. Therefore, the process will becompleted when the following condition is satisfied.
x= U
1+ U2+ ... + U
k+1. (3)
Note that only when the system encounters Poisson failures, U
1 and U
k+1 will still have
the same exponential distribution as those Uis in between. It is this fact that makes the
process completion time r.v.s independent of each other. When a general i.i.d. r.v. models
the interruption process or there exists autocorrelation in the times to failure stream, the
process completion times will become autocorrelated. This is the very reason why the pro-
cess completion time analysis imposes bigger challenge when the interruption process is not
Poisson.
In our model, a 2-state H2 r.v. governs the times to failure, which implies that during any
up time the failure process will be in either of the states i, j = 1, 2. If an idle period starts,
the failure process stays in the same state until another job comes. Therefore, we can use
the semi-Markov approach, since the state transitions are imbedded in service completions,
i.e., departure instants.
To compute the mean waiting time in this system, we need to compute the steady-state
probability of having ncustomers in the queue and failure process in state i, namely,i(n).
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It is clear that(n) =1(n) + 2(n) gives the steady-state probability of havingn customers
in the queue since Poisson arrivals see the time averages (PASTA property).
Computing i(n) requires the transition probability of the queue length process,pi,j(n),
i.e., the probability of havingn arrivals during the process completion time of a job, if failure
process ends in state j given that it started in state i. To this end, we make use ofqi,j(k, x)
defined as the conditional probability that there happen k failures during the processing
time x and failure process ends in state j, given that it was in state i when the job seized
the machine, i, j = 1, 2 and k = 0, 1, 2,... . Additionally, we introduce an intermediate
probability,p(k, n), which is the probability of havingn Poisson arrivals when k failures are
observed. Hence,
pi,j(n) =k=0
qi,j(k, x)p(k, n), (4)
where qi,j(k, x) rapidly tends to 0 as k increases.
Note that we have a Markov-chain {Ss, Nq, s= 1, 2, q= 0, 1, 2,...} whereSsis the state ofthe failure process andNq is the number of jobs left behind in the queue at departure epochs,
which can be truncated at a sufficiently large Kvalue (guaranteed to exist for systems with
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P =
p1,1(0) p2,1(0) p1,1(0) p2,1(0) 0 0 . . . 0
p1,2(0) p2,2(0) p1,2(0) p2,2(0) 0 0 . . . 0
p1,1(1) p2,1(1) p1,1(1) p2,1(1) p1,1(0) p2,1(0) 0 0 . 0
p1,2(1) p2,2(1) p1,2(1) p2,2(1) p1,2(0) p2,2(0) 0 0 . 0
. . . . . . . . . .
p1,1(K) p2,1(K) p1,1(K) p2,1(K) . . . . p1,1(1) p2,1(1)
p1,2(K) p2,2(K) p1,2(K) p2,2(K) . . . . p1,2(1) p2,2(1)
.
If 2(K+ 1) 1 0 denotes the initial probability vector of having n customers in thequeue while failure process is in state i at time 0 (which can be assumed to be 0 =
[p, 1 p, 0, ..., 0, 0]T, since the choice does not have an impact on ),
= limN
PN0, (5)
subject toKn=0
1(n) + 2(n) = 1,
hence, (n) = 1(n) +2(n), from which the mean number of customers in the queue,
and via Littles formula, the mean waiting time can be computed easily. For computational
purposes, matrix multiplication ofPis performed sufficiently many times. In the numerical
examples we considered,K= 100 and N= 2000 were more than sufficient.
Federgruen and Green (1988) use matrix calculations to compute qi,j(k, x), and then
express pi,j(n) in terms of infinite sums to avoid numerical integration. Finally, they em-
ploy an aggregation/disaggregation procedure to obtain i(n), which is reported to be time
consuming for certain problems.
In this paper, we propose an alternative technique to computeqi,j(k, x),pi,j(n) andi(n).
First of all, it is easy to check that
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q1,1(0, x) = e1x,
q1,2(0, x) = 0,q2,1(0, x) = 0,
q2,2(0, x) = e2x, (6)
and fork >0,
q1,1(k, x) = x0
(p q1,1(k 1, x u) + (1 p) q2,1(k 1, x u))1e1udu,
q1,2(k, x) = x0
(p q1,2(k 1, x u) + (1 p) q2,2(k 1, x u))1e1udu,
q2,1(k, x) = x0
(p q1,1(k 1, x u) + (1 p) q2,1(k 1, x u))2e2udu,
q2,2(k, x) = x0
(p q1,2(k 1, x u) + (1 p) q2,2(k 1, x u))2e2udu. (7)
It appears to be difficult to obtainqi,j(k, x) using Eq. (7). We suggest using their Laplace
transforms, which can be inverted to arriveqi,j(k, x). The Laplace transforms of interest are,
q1,1(0, s) = 1/(1+ s),
q1,2(0, s) = 0,
q2,1(0, s) = 0,
q2,2(0, s) = 1/(2+ s), (8)
and fork >0,
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q1,1(k, s) = gR(s)k1 p 1
1+ s
1
1+ s,
q1,2(k, s) = g
R(s)k1
(1 p) 11+ s
1
2+ s,
q2,1(k, s) = gR(s)k1 p 2
2+ s
1
1+ s,
q2,2(k, s) = gR(s)k1 (1 p) 2
2+ s
1
2+ s, (9)
where gR
(s) is given in Eq. (1). We numerically invert Eq.s (8) and (9) using the technique
proposed by Jagerman (1982) and compute qi,j(k, x).
Now we will computep(k, n), which is the probability of havingnPoisson arrivals when k
failures happen during processing timex. LetDk define the total length ofk repair times the
machine undergoes while servicing a job ifkfailures are observed. ThenC(x, k) =x+Dk will
be the total time (effective service time) a job spends on the machine. Denoting B(t,x,k)
as the distribution function ofC(x, k) (t representing the process completion time), we can
write its Laplace-Stieltjes transform B(s,x,k) as
B(s,x,k) = es x[fD
(s)]k, (10)
where fD
(s) is the Laplace transform of the down/repair time density function. Then p(k, n)
is given by
p(k, n) =
0
( C(x, k))n
n! eC(x,k)dB(t,x,k). (11)
Ifvdenotes the number of arrivals during C(x, k), its z-transform, namely V(z) =
n=0 P[v=
n]zn, can be expressed in terms ofB(s,x,k) (Kleinrock, 1975, page 184):
V(z) = B((1 z), x , k). (12)
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Since (Kleinrock, 1975, page 336)
p(k, n) =
1
2iCz
1n
V(z)dz, (13)
where i =1, using the fact that the path is a circle around the origin given by z =
rei, r
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3 Numerical Examples for M/PCT/1 Queues
In order to asses the efficacy of the proposed method, we have used a simple TES+ process to
model autocorrelated times to failure with marginal exponential distribution characterizedby a parameter triplet (L,R,
X) (Jagerman and Melamed, 1992a, 1992b). While
X is the
rate of the exponentially distributed times to failure, [L, R) subject to 0.5 L R
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is x= 5, and the failure rate is X
= 0.8. Using the approximating H2 times to failure r.v.,
we have obtained the analytical results wapx using Eq.(5) and compared with its reference
counterpart, wsim, by computing the error metric
( wapx, wsim) =wapx wsim
wsim 100. (18)
In Tables 1 - 3, the first column displays the TES+ process used as the autocorrelated
interruption process. In column 2, we present the lag-1 autocorrelation function of the
process completion time r.v., C(x)
(1), also found from simulation. Column 3 lists the mean
waiting times estimated from simulation with their 95% confidence intervals given beneath
each. Finally, in the last column we present our analytical approach with its approximation
error computed using Eq. (18). A comparison of the approximation method to the reference
values obtained by simulation reveals the following facts.
Higher positive autocorrelation levels in the times to failure process induces morepositive autocorrelation in the process completion time r.v.
Table 1 displays the results where repair times are drawn from a Uniform(0.5,1) dis-
tribution yielding an overall 30% system down time probability. This table shows
the impact of positive autocorrelation most dramatically. The mean waiting time in-
creases by 125% from the i.i.d Poisson failures case (TES+ A) to the most positively
autocorrelated case (TES+ D). The renewal approximation performs consistently well
in predicting the mean waiting times of the original problem.
Table 2 displays the results where repair times are drawn from a Uniform(0.34,0.5)
distribution yielding an overall 20% system down time probability. The mean waiting
time increases by 54% from the i.i.d Poisson failures case (TES+ A) to the most posi-
tively autocorrelated case (TES+ D). The renewal approximation performs consistently
well in predicting the mean waiting times of the original problem.
Table 3 displays the results where repair times are drawn from a Uniform(0.1,0.26)distribution yielding an overall 10% system down time probability. This table shows
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the cases where the impact of autocorrelation is mitigated. Yet, the mean waiting time
increases by 13% from the i.i.d Poisson failures case (TES+ A) to the most positivey
autocorrelated case (TES+ D). The renewal approximation performs consistently well
in predicting the mean waiting times of the original problem.
The results indicate that if the repair times are lengthy, the existence of positive auto-correlation in the interruption process increases the mean waiting time tremendously.
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Simulation Analytical
TES+ C(x)
(1) W W
A 0.00 16.15 16.58
(+/- 0.62) (2.68%)
B 0.24 17.86 17.93
(+/- 0.1) (0.38%)
C 0.45 21.17 21.46
(+/- 0.13) (1.38%)
D 0.62 36.72 36.91
(+/- 0.78) (0.76%)
Table 1: Mean waiting times in M/G/1 Systems with = 0.1, TES+ times to failure process
and fD
(d) = Uniform(0.5,1), with d1= 0.75, d2= 0.5833, and 30% down time
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Simulation Analytical
TES+ C(x)
(1) W W
A 0.00 13.576 13.58
(+/- 0.393) (-0.01%)
B 0.24 14.12 14.08
(+/- 0.03) (-0.27%)
C 0.45 15.25 15.39
(+/- 0.08) (0.92%)
D 0.62 20.48 20.71
(+/- 0.11) (1.12%)
Table 2: Mean waiting times in M/G/1 Systems with = 0.11976, TES+ times to failure
process and fD
(d) = Uniform(0.34,0.5), with d1 = 0.42, d2 = 0.17854, and 20% down time
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Simulation Analytical
TES+ C(x)
(1) W W
A 0.00 11.378 11.493
(+/- 0.284) (1.01%)
B 0.24 11.58 11.61
(+/- 0.06) (0.27%)
C 0.44 11.83 11.89
(+/- 0.04) (0.52%)
D 0.61 12.82 12.94
(+/- 0.06) (0.92%)
Table 3: Mean waiting times in M/G/1 Systems with = 0.1399, TES+ times to failure
process and fD
(d) = Uniform(0.1,0.26), with d1 = 0.18, d2 = 0.03454, and 10% down time
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4 Approximations for the GI/PCT/1 Queues
The exact analysis presented in Section 2 for systems experiencing H2 times to failure de-
pends on the assumption of Poisson customer/job arrivals. The main difficulty is our inabilityof expressing B(s, x) explicitly. To investigate the GI/PCT/1 queues, there are two alter-
natives. One can directly collect the autocorrelated process completion time samples, and
analyzing the data, can approximate it with an appropriate renewal r.v. in line with the
techniques proposed by Jagerman et al. (2004) and Balcoglu, Jagerman and Altok (2005).
However, this approach does not leave space to conduct sensitivity analysis in which differ-
ent down time distributions can be assumed. For each different down time density function
fD
(d), one will have to collect different process completion time data sets, which may not
even exist. On the other hand, if the interruption process is assumed to stay the same, since
qi,j(k, x) is computed independent of the down time distribution, the process completion
time analysis provides a better alternative.
To extend the analysis presented in Section 2, we will first construct an approximating
M/G/1queue with i.i.d. service time r.v. with a Laplace transform of b(s) approximating
theM/PCT/1queue with autocorrelated service time due to H2 times to failure. Note that
the Poisson customer arrival rate of the approximatedM/PCT/1will be equal to the renewal
customer arrival rate in the GI/PCT/1queue. In the approximatingM/G/1queue, the P-K
transform equation (Kleinrock, 1975, page 194) would hold:
Q(z) = b((1 z))(1 )(1 z)b((1 z)) z, (19)
which can be re-written as
b(z) =z
Q(z
)
Q(z
) (1 )(1 z
), (20)
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where Q(z) =
n=0 (n) zn is the z-transform of the queue length process, which will
be equal to that of the M/PCT/1 queue with H2 failure process. Then after a change of
variable, one can express b(s) in terms ofQ(s) as
b(s) =
n=0(s
)n+1(n)
n=0(s
)n(n) (1)s
, (21)
where (n) is found via Eq. (5).
It is clear that theGI/G/1queue that uses the sameb(s) will approximate theGI/PCT/1
queue with H2 times to failure. This will be an additional error source with respect to the
GI/PCT/1 queue with autocorrelated times to failure. For numerical computations, Eq.
(21) is not easy to use, either. However, one can compute the moments of b(s) and choose
phase-type (PH) r.v.s that match the first two moments exactly, and the third moment with
the least error. The rationale behind incorporating the third moment is due to Balcoglu,
Jagerman and Altok (2005), who demonstrate that two-moment matching techniques could
incur big errors. Definitely, this will be the third source of approximation error with respect
to theGI/PCT/1queue with autocorrelated times to failure. However, if the overall error is
small, the approximation can be considered efficient and accurate. Accordingly, ifb(s) yields
a squared-coefficient of variation, c2 < 1, we choose a Generalized-Erlang r.v., GE(,p,k),
which has the following Laplace transform:
b(s) = (1 p) + s
+ p(
+ s)k. (22)
In casec2 >1, we will choose an H2 r.v., H2(p, 1, 2), whose Laplace transform is presented
in Eq. (1). For the Poisson interruption process (TES+ A) the analysis is still exact since
we have Eq. (16).
In the numerical examples presented in Tables 5 and 6, the down times are uniformly
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distributed over (0.5,1), since this is the case in which the impact of positively autocorrelated
interruption process is felt most dramatically. Here, independent of which TES+ process is
used, the first moment of the process completion time is E[C(x)] = 8, since autocorrelation
does not reveal its impact on this statistics. The first column in Table 4 lists the TES+
interruption processes used, which change the second and third moments of the process
completion r.v. that are displayed in columns 2 and 3, respectively. The distributions given
in column 4 match the first two moments, exactly, and approach the third moment with the
possible minimum error. Their third moments are listed in the last column.
TES+ E[C(x)2] E[C(x)3] PH-Distribution E[C(x)3PH
]
B 71.71 730.5 GE(= 1.115, p= 0.99, k = 9) 707.6
C 85.9 1325.5 GE(= 0.372, p= 0.987, k= 3) 1154
D 148.15 6478.7 H2(1= 0.13113, 2= 0.0286, p= 0.9863) 6125.37
Table 4: Phase-type Distributions approximating PCT in the G/PCT/1 queue
Tables 5 and 6 have the same structure as Tables 1 - 3. The analytical solution presentedin the last column is found via the exact mean waiting time computation in the GI/GI/1
queue (we refer the interested reader to Riordan (1962) pages 50-52 for the details).
A comparison of the approximation methods to the reference values obtained by simula-
tion reveals the following facts.
Table 5 displays the results where the job/customer arrival process is a smooth process
with GE(0.2,1,2) interarrival times (see Eq. 22). Although the mean waiting times are
shorter than the ones presented in Table 1, this measure increases by 249% from the
i.i.d Poisson failures case (TES+ A) to the most positively autocorrelated case (TES+
D). TheGI/G/1 approximation performs consistently well, however for the cases with
TES+ B and TES+ D, the error is bigger than their corresponding cases in Table 1. The
two additional error sources on top of the renewal approximation become influential
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when the job/customer arrival process has a squared-coefficient of variation less than
1.
Table 6 displays the results where the job/customer arrival process is a bursty process
with interarrival times following H2 with parameters (0.88889, 0.2, 0.02) (see Eq. 1),
with a squared coefficient of variation equal to 5. The mean waiting time increases by
31% from the i.i.d Poisson failures case (TES+ A) to the most positively autocorrelated
case (TES+ D). In these cases, the high variability in the job/customer arrival process
becomes the more dominant factor on the mean waiting time.
Although it has two additional sources of error, the results indicate that, the GI/G/1
approximation can be used to analyze theGI/P CT/1 queues with autocorrelated times
to failure.
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Simulation Analytical
TES+ C(x)
(1) W W
A 0.00 7.68 7.78
(+/- 0.18) (1.25%)
B 0.24 8.70 9.05
(+/- 0.24) (4.07%)
C 0.45 12.37 12.47
(+/- 0.56) (0.8%)
D 0.62 26.82 27.78(+/- 2.38) (3.58%)
Table 5: Mean waiting times inGI/G/1 Systems with GE(0.2,1,2) interarrival times, TES+
times to failure and andfD
(d) = Uniform(0.5,1), with d1 = 0.75, d2 = 0.5833, and 30% down
time
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Simulation Analytical
TES+ C(x)
(1) W W
A 0.00 78.854 78.63
(+/- 0.84) (-0.28%)
B 0.24 80.958 80.25
(+/- 1.59) (-0.88%)
C 0.45 83.172 84.43
(+/- 1.97) (1.51%)
D 0.62 103.24 101.79(+/- 3.59) (-1.4%)
Table 6: Mean waiting times in GI/G/1 Systems with H2(0.88889,0.2,0.02) interarrival
times, TES+ times to failure and fD
(d) = Uniform(0.5,1), with d1 = 0.75, d2 = 0.5833, and
30% down time
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5 Conclusions
In this paper, we have developed a process completion time (PCT) analysis for a workstation
that encounters autocorrelated times to failure with i.i.d. repair times. To do this, weemployed a three-parameter renewal approximation. The approximating times to failure r.v.
is of 2-state Hyper-exponential (H2) type. When the customer arrival process is Poisson,
an exact computation of the mean waiting time in a single server queue with H2 times to
failure is achieved as shown in Section 2. In the case of renewal arrivals, an approximate yet
accurate approach is proposed in Section 4. This approximation works on the principle of
computing the mean waiting time in a GI/G/1 queue by making use of an M/G/1 queue
that has the same service time distribution function as that of the former.
Numerical examples demonstrate that the autocorrelation in times to failure should be
incorporated in the analysis. Especially, in queues receiving smooth arrivals, i.e., with an
interarrival time squared-coefficient of variation less than 1, increased levels of positive au-
tocorrelation degrade system performance sharply. In all the cases considered, the three-
parameter renewal approximation proved to be accurate in capturing the behavior of the
autocorrelated interruption processes.
Acknowledgements
This work was supported in part by NSF Grants DMI-9812858 and DMI-0085659.
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