Analysis of a two-class FCFS queueing system with interclass correlation

TLDR: The major aim of the paper is to estimate the impact of the interclass correlation in the arrival stream on the queueing performance of the system, in terms of the ( average) number of customers in the system and the (average) customer delay and customer waiting time.

Abstract: This paper considers a discrete-time queueing system with one server and two classes of customers. All arriving customers are accommodated in one queue, and are served in a First-Come-First-Served order, regardless of their classes. The total numbers of arrivals during consecutive time slots are i.i.d. random variables with arbitrary distribution. The classes of consecutively arriving customers, however, are correlated in a Markovian way, i.e., the probability that a customer belongs to a class depends on the class of the previously arrived customer. Service-time distributions are assumed to be general but class-dependent. We use probability generating functions to study the system analytically. The major aim of the paper is to estimate the impact of the interclass correlation in the arrival stream on the queueing performance of the system, in terms of the (average) number of customers in the system and the (average) customer delay and customer waiting time.

Figures

Fig. 4. Mean system content E[s] versus load ρ for various values of the interclass correlation γ

Fig. 7. Mean waiting time E[w] versus the fraction tA of class-A customers for various values of the load ρ

Fig. 1. Two-state Markov chain of the customer classes

Fig. 6.Mean system content E[s] versus the fraction tA of class-A customers for various values of the load ρ

Fig. 5. Mean system content E[s] versus interclass correlation γ for various values of the load ρ

Fig. 3. Relationship between uk and uk+1 when uk = 0

Full text

Analysis of a two-class FCFS queueing system
with interclass correlation
Herwig Bruneel, Tom Maertens, Bart Steyaert, Dieter Claeys, Dieter Fiems,
and Joris Walraevens
Ghent University,
Department of Telecommunications and Information Processing,
SMACS Research Group,
Sint-Pietersnieuwstraat 41,
B-9000 Ghent, Belgium
{hb,tmaerten,bs,dclaeys,df,jw}@telin.UGent.be
Abstract. This paper considers a discrete-time queueing system with
one server and two classes of customers. All arriving customers are ac-
commodated in one queue, and are served in a First-Come-First-Served
order, regardless of their classes. The total numbers of arrivals during
consecutive time slots are i.i.d. random variables with arbitrary distri-
bution. The classes of consecutively arriving customers, however, are
correlated in a Markovian way, i.e., the probability that a customer be-
longs to a class depends on the class of th e previously arrived customer.
Service-time distributions are assumed to be general but class-dependent.
We use probab ility generating functions to study th e system analytically.
The major aim of the paper is to estimate the impact of the interclass
correlation in the arrival stream on the queueing performance of th e sys-
tem, in terms of the (average) number of customers in the system and
the (average) customer delay and customer waiting t ime.
1 Introduction
Various ty pes of scheduling disciplines have been investigated within the context
of multi-class q ueue ing systems. We mention, among others, priority scheduling
(see, e.g., [4, 8, 11, 13, 15]), weighted fair queueing (WFQ) (see, e.g., [14, 17]),
random order of service (ROS) (see, e .g., [1 , 3, 10]), and generalized processor
sharing (GPS) (see, e.g., [9,12,16]). Strangely enough, only few re sults have been
derived fo r multi-class First-Come-First-Served (FCFS) systems, i.e., queueing
systems in which the customers of different classes are accommodated in one
queue and served in their order of arrival, irrespective of the classes they belong
to (a recent paper is [5]). The present paper presents the analysis of a discrete-
time model that fits in this category.
In c lassical multi-class queueing models, furthermore, it is generally assumed
that the different cla sses occur randomly and independently in the arrival stream
of customers (this is also the case in [5]). In this pap er, however, we explicitly
wish to exa mine the effect of so-called interclass correlation (or class clustering)

2 Herwig Bruneel et al.
in the arrival process. Specifically, we are interested to know whether the degre e
to which customers of the same class have the tendency to arrive (and be served)
closely together (i.e., back-to-back), or, co nversely, the degree to which such
customers have the tendency to be spr ead in time and mixed with c ustomers
of the other class, has a substantial impact on the performance of a two-class
FCFS queueing system. In order to do so, we superimpose a two-state Markovian
inte rclass correlation model (with arbitrary transition probabilities) on top of a
regular general independent arrival process model for the aggregated customer
stream. Servic e -time distributions are class-dependent but completely general. It
is clear that the interclass correlation between consecutive customers can also be
viewed as a form of non-independence between service times. One application
of this queueing model is obvious: the two customers classes can model, for
example, voice and data packets in a heterogeneo us telecommunication system.
It is common knowledge that data packets are significa ntly larger than voice
packets. Then it is easy to see that if data packets have the tendency to arrive in
clusters, the p erformance of the system may be degrade d severely (with respect
to voice pa ckets). In this paper, we measure this degradation.
We first derive the probability generating function (pg f) of the total number
of customers in the system at customer depa rture times. From this result, we
can easily obtain the corresponding pgf valid at arbitrary slot boundaries. Var-
ious performance measures of practical use, such as the mean system content,
the mean cus tomer delay and the mean customer waiting time, can be ea sily
derived from these pgf’s by applying the moment-generating property of pgf’s
and by using Little’s law. The resulting fo rmulas and a number of numerical
examples reveal that the system under study can exhibit two types of stochas-
tic equilibrium, depending on the values of the system parameters: a “strong”
equilibrium in w hich b oth customer classes individually generate less work than
the system can handle (dur ing periods where only such customers arrive), and a
“compensated” type of equilibrium whereby one customer class creates overload
situations which ar e compensated by stro ng under-load periods generated by the
other customer class. In the latter case, our results clearly demonstra te the cru-
cial importance of the amount of interclass correlation on the usual performance
parameters of the system.
The outline of this pape r is as follows. In Section 2, we describe the math-
ematical model. Section 3 first presents an analysis of the total number of cus-
tomers in the system at customer departure times; next, the pgf of the system
content at random slot boundaries is derived from this result. We discuss the
results, both conceptually and quantitatively, in Section 4. Some conclusions are
drawn in Section 5.
2 Mathematical model
We consider a discrete-time queueing system with infinite waiting room, one
server, and two classes of customers, named A and B. As in all discrete-time
models, the time axis is divided into fixed-length intervals referred to as slots in

Lecture Notes in Compu ter Science 3
the sequel. New customers may enter the system at any given (continuous) point
on the time axis , but services are synchronized to (i.e., can only start and end
at) slot boundaries . Customers are s erved in their order of arrival, regardless of
the class they b elong to. We call this s ervice discipline “global FCFS”.
1− α
β
1− β
α
A B
Fig. 1. Two-state Markov chain of the customer classes
The arrival process of new customers in the system is characteriz e d in two
steps. First, we model the total (aggre gated) arrival stream of new customers by
means of a sequence of i.i.d. non-negative discrete random variables (denoting
the numbers of arrivals in c onsecutive slots) with common probability generating
function (pgf) E(z). The (total) mean number o f arrivals per slot, in the sequel
referred to as the (total) mean arrival rate, is given by λ , E
0
(1). Next, we
describe the o c currence of the two classes in the sequence of the consecutively
arriving customers. In this study, we assume that both classes of customers ac-
count for pa rt of the total load of the system, i.e., both customer classes are
“mixed” in the arrival stream, but there may be so me degree of “class cluster-
ing” in the arrival process, i.e., customers of any given class may (or may not)
have a tendency to “ arrive back-to-back”. Mathematically, this means that the
classes of two consecutive customers may be non-independent. Specifically, we
assume a first-order Markovian type of corr elation between the classes of two
consecutively arriving customers, w hich basically means that the probability
that the next customer belongs to a given class depends on the class of the pr e-
viously arrived c ustomer. Let t
k
denote the class of customer k. The transition
probabilities of the Markov chain that determines the class o f the consecutively
arriving customers are then defined as (see Fig. 1)
Prob[t
k+1
= A | t
k
= A] , α , (1)
Prob[t
k+1
= B | t
k
= A] , 1 − α , (2)
Prob[t
k+1
= A | t
k
= B] , 1 − β , (3)
Prob[t
k+1
= B | t
k
= B] , β . (4)
It is well known that for a two-s tate Markov chain of this type, the steady-state
probabilities t
A
and t
B
of finding the chain in state A and B are given by
t
A
, lim
k→∞
Prob[t
k
= A] =
1 − β
2 − α − β
(5)

4 Herwig Bruneel et al.
and
t
B
, lim
k→∞
Prob[t
k
= B] =
1 − α
2 − α − β
,
(6)
respectively (see, e.g., [2]). The quantitie s t
A
and t
B
can be interpreted as the
fractions of class A and class B customers in the ar rival str e am. The (steady-
state) correlation coefficient of the Markov chain, i.e., the amount of correlation
between the clas ses of two consecutively ar riving customers (in the steady state),
is given by
γ , lim
k→∞
E[t
k
t
k+1
] − E[t
k
]E[t
k+1
]
p
var[t
k
] var[t
k+1
]
= α + β − 1 . (7)
We will indicate the parameter γ (−1 ≤ γ ≤ +1) as the interclass correlation in
the sequel. Positive values of γ correspond to a situation whereby the customer s
of any given class have a tendency to cluster, while neg ative values of γ refer
to arrival streams in which the customers of classes A a nd B have a tendency
to alternate, i.e., b e mixed more strongly. The case where γ = 0, of cour se,
corres ponds to the classical assumption that classe s of subsequent customers are
independent.
The service pr ocess of the system is characterized by attaching to e ach cus-
tomer a corresponding service time, which indicates the numbe r of time slots
required to give complete service to the customer at hand. The service times of
customers are class-dependent and are modelled as a sequence of independent
positive discrete random variables with pgf’s A(z) and B(z). The corresponding
mean values are given by µ
A
, A
0
(1) and µ
B
, B
0
(1) for customers of class A
and B, respectively.
3 System analysis
3.1 System equations at customer departure times
Let u
k
denote the total system content, i.e ., the total number of customers
present in the sy stem just after the service completion of the k-th customer,
and, as before, let t
k
indicate the class customer k belongs to. Then, as a con-
sequence of all the model assumptions in Section 2, the co uple (t
k
, u
k
) forms a
Markovian state description of the system (at customer departure times).
The state transitions of the quantities {t
k
} are governed by the Eqs. (1)-(4),
whereas for the quantities {u
k
}, the following re c ursive system equations can be
established (see Figs. 2 and 3):
u
k+1
=

u
k
− 1 + g
k+1
if u
k
> 0
f
k+1
+ g
k+1
if u
k
= 0
. (8)
Here, the quantity g
k+1
is defined as the number of ar rivals in the system during
the service time of customer k +1 , while f
k+1
indicates the numb er of customers
arriving after customer k + 1 in its arrival slot.

Lecture Notes in Compu ter Science 5
u
k
u
k+1
{
g
k+1
departureof
customerk
departureof
customerk+1
>0
time
Fig. 2. Relationship between u
k
and u
k+1
when u
k
> 0
time
u
k
u
k+1
{
g
k+1
departureof
customerk
departureof
customerk+1
arrivalof
customerk+1
=0
{
f
k+1
Fig. 3. Relationship between u
k
and u
k+1
when u
k
= 0
It is easily seen that the pgf o f f
k+1
is given by the pg f of the number of
additional arrivals in a slot with at least one arrival, i.e.,
F (z) , E[z
f
k+1
] =
E(z) − E(0)
z[1 − E(0)]
, (9)
regardless of the class of customer k + 1. The distr ibution of the quantity g
k+1
,
however, does depend on the class of customer k + 1. We have
G
A
(z) , E[z
g
k+1
| t
k+1
= A] = A(E(z)) , (10)
G
B
(z) , E[z
g
k+1
| t
k+1
= B] = B(E(z)) . (11)
3.2 System content at customer departure times
Let us assume that the queueing system at hand is stable, i.e., that the stability
condition is fulfilled. Intuitively, it is not difficult to see that the system is stable
if and only if the average amount of work entering the system per slot is strictly
less than 1, i.e., if and only if λE[c] < 1, w ith E[c] the average service time of
an arbitrary c ustomer. Expr essed in the basic parameters of o ur system, this is
equivalent to the condition
λ(t
A
µ
A
+ t
B
µ
B
) < 1 , (12)
where the quantities t
A
and t
B
are the steady-state probabilities of the arrival
Markov chain (s ee Eqs. (5) and (6)). Assuming this condition fulfilled, we define

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Analysis of a two-class fcfs queueing system with interclass correlation" ?

This paper considers a discrete-time queueing system with one server and two classes of customers. The authors use probability generating functions to study the system analytically. The major aim of the paper is to estimate the impact of the interclass correlation in the arrival stream on the queueing performance of the system, in terms of the ( average ) number of customers in the system and the ( average ) customer delay and customer waiting time. 

Q2. What are the future works in "Analysis of a two-class fcfs queueing system with interclass correlation" ?

If, however, tA increases further ( while the total load ρ remains constant ), the system receives considerably less customers ( for the same amount of work ), which explains the decreasing system content in the interval 0. 1 ≤ tA ≤ 1. 

Q3. What is the effect of interclass correlation on the system?

Positive interclass correlation appears to be very detrimental for the performance of the system, whereas negative interclass correlation has a very moderate positive effect on the performance. 

Q4. How can the authors obtain the pgf of the system content at random slot boundaries?

Various performance measures of practical use, such as the mean system content, the mean customer delay and the mean customer waiting time, can be easily derived from these pgf’s by applying the moment-generating property of pgf’s and by using Little’s law. 

Q5. What is the effect of the tA increase on the system?

however, tA increases further (while the total load ρ remains constant), the system receives considerably less customers (for the same amount of work), which explains the decreasing system content in the interval 0.1 ≤ tA ≤ 1. 

Q6. What is the limiting value of E[s]?

It is not surprising that E[s] goes to infinity as ρ approaches its limiting value 1, dictated by the stability condition of the system. 

Q7. What is the first example of a Poisson arrival?

In a first example, the authors assume Poisson arrivals (i.e., E(z) = eλ(z−1)), equal fractions of both classes of customers in the arrival stream (i.e., tA = tB = 0.5), geometrically distributed service times for both classes, i.e.,X(z) = zµX + (1− µX)z , (31)with X ∈ {A,B}, and with µA = 8 and µB = 2. 

Q8. What is the mean system content for tA?

Fig. 6 reveals that, for any given value of the total load ρ, the mean system content increases as a function of tA for “low” values of tA (more or less in the interval 0 ≤ tA ≤ 0.1), then reaches a maximum value for tA somewhere around 0.1, and, finally decreases monotonically in the interval 0.1 ≤ tA ≤ 1. 

Q9. What is the effect of the mean waiting time on the system?

More specifically, it can be observed that the mean waiting times also increase for “low” values of tA to reach a maximum value and then decrease for “higher” values of tA. 

Q10. How do the authors model the arrival stream of new customers?

the authors model the total (aggregated) arrival stream of new customers by means of a sequence of i.i.d. non-negative discrete random variables (denoting the numbers of arrivals in consecutive slots) with common probability generating function (pgf) E(z). 

Q11. What are some of the other scheduling disciplines that have been investigated in the context of multi-class?

The authors mention, among others, priority scheduling (see, e.g., [4, 8, 11, 13, 15]), weighted fair queueing (WFQ) (see, e.g., [14, 17]), random order of service (ROS) (see, e.g., [1, 3, 10]), and generalized processor sharing (GPS) (see, e.g., [9,12,16]). 

Q12. How can the authors determine the stability of the complex z-plane?

it can be shown by means of Rouché’s theorem from complex analysis [2,7] that the denominator of Eq. (22) has exactly two zeroes inside the closed unit disk of the complex z-plane, one of which is equal to 1, as soon as the stability condition (12) is fulfilled. 

Q13. What is the simplest way to determine unknowns?

The unknowns can be determined, in general, by invoking thewell-known property that pgf’s such as P (z) are bounded inside the closed unit disk {z : |z| ≤ 1} of the complex z-plane, at least when the stability condition (12) of the queueing system is met (only in such a case their analysis was justified and P (z) can be viewed as a legitimate pgf). 

Q14. What is the stability condition of the arrival stream?

In the latter case, labelled as the “compensated” equilibrium, (global) stability is assured because although the queue size builds up during A-sequences (because, on average, more work arrives than the server can perform), it decreases again during B-sequences (when much less work enters than the server can execute). 

Q15. What is the pgf of the system content at random slot boundaries?

New customers may enter the system at any given (continuous) point on the time axis, but services are synchronized to (i.e., can only start and end at) slot boundaries. 

Q16. What are the recursive system equations for the quantities tk?

The state transitions of the quantities {tk} are governed by the Eqs. (1)-(4), whereas for the quantities {uk}, the following recursive system equations can be established (see Figs. 2 and 3):uk+1 ={uk − 1 + gk+1 if uk > 0 fk+1 + gk+1 if uk = 0 . (8)Here, the quantity gk+1 is defined as the number of arrivals in the system during the service time of customer k+1, while fk+1 indicates the number of customers arriving after customer k + 1 in its arrival slot. 

Q17. What is the reason why the system content is so small?

this can be attributed to the fact that the waiting time reflects the unfinished work in the system (at the arrival instant of a customer), while the system content indicates the number of customers in the system, whereby all customers contribute identically, irrespective of their servicetime, i.e., irrespective of the amount of work they represent.