Analysis Queue With Autocorrelated Times to Failures (1)

8/12/2019 Analysis Queue With Autocorrelated Times to Failures (1)

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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 =

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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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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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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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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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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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Analysis Queue With Autocorrelated Times to Failures (1)