[ieee 2013 ieee 21st international symposium on modelling, analysis & simulation of computer and...

Networks of Order Independent Queues withSignals

Thu-Ha DAO-THI

PRiSM

Univ. Versailles St-Quentin, CNRS

Versailles, France

Email: [email protected]

Jean-Michel FOURNEAU

PRiSM

Univ. Versailles St-Quentin, CNRS

Versailles, France


Minh-Anh TRAN

LACL

Univ. Paris Est Créteil

Créteil, France


Abstract—We study the steady-state distribution of networksof order independent queues with negative signals which deletecustomers. An Order Independent queue is defined by a servicerate which is independent on the order of the customers in thequeue. Such an abstract discipline may be used to model complexblocking mechanism (for instance the Multiserver Station withConcurrent Classes of Customers). Order independent queuesare in general neither symmetric nor reversible. We prove that,under usual assumptions on the arrivals, the services and therouting of customers, such a network of queues with signals hasa steady-state distribution with product form solution. The proofis based on the quasi-reversibility of the queues. We also presentsome examples of application for this new analytical result.

I. INTRODUCTION

Order Independent queues (OI queues in that follows) are an

abstract way to describe a large family of queuing disciplines

which allow a potential blocking and complex contention

mechanisms between customers. They also model simpler

queueing disciplines. These queues are quasi-reversible. Thus,

one can build networks of such queues with a Markov routing

which have a product form steady-state distribution. These

queues may exhibit complex behaviors but they have not been

extensively studied as a set of consistent disciplines which

share some structural properties (see chapter 2 in [1] for an

exception). In this paper we show how we can add signals to

Order Independent queues and still have a steady-state solution

which has a product form. We consider two types of signal: in

a single queue, a negative customer which deletes a positive

customer if there is any in the queue, and in the network

model, a negative signal which deletes a positive customer

and can have instantaneous movement between queues after

successful cancellation.

Order Independent queues were defined as follows in [1],

chapter 2 (see also [2]):

1) The state of the system is described by a vector of types

of customer.

2) The rate of service for customer in position i depends

only on the composition of the queue up to position i.Of course i is included.

3) For any state, the rate of service is the same if we reorder

the customers in the queue with any permutation.

4) At each state except the empty queue, the rate of service

is positive

A more formal definition will be given in the next section.

Many well-known queueing discipline are order independent.

Clearly, First Come First Served with one class of customers

is order independent. Infinite Serves queue, M/M/K queue

and Processor Sharing (PS) with multiple classes are also

order independent while LCFS is not. More complex queueing

disciplines may be also represented as well. Indeed, some

parts of the service capabilities may be wasted for some states

and this allows to model the blocking of customers due to

some contention on multiple resources. A typical example

of such a discipline is the Multiserver Station with Concur-

rent Classes of Customers (MSCCC in the following) and

another example is the Multi Server Hierarchical Concurrent

Customers (MSHCC). MSCCC was first considered in [3]

while the first results came from a preliminary version of

[4]. The authors considered a multiple bus multiprocessor

system. A memory request emitted by a processor needs a

bus and a memory connected to the bus. Thus a customer

(i.e. a request from a processor) requires two resources of

distinct types: a bus and a memory. In [4], the local balance

property has been proved for a configuration with two busses.

Such a property was numerically checked with Generalised

Stochastic Petri nets for systems with up to five buses and

less than 10 processors. These results opened the way to more

complex results on arbitrary configurations. Le Boudec defined

in [3] the MSCCC queue as follows: the station contains Bidentical exponential servers, with constant service rate. At a

station, the classes of customers are sorted into M concurrent

groups ; the discipline of service is on a First Come First

Served basis, but two customers of the same group cannot

be served simultaneously. He showed that such stations can

be inserted in BCMP networks preserving the product form

solution. Formally, one can define a MSCCC queue as a station

consisting of B parallel identical exponential servers. The

customers belong to K groups. Parameter n represents the

amount of parallelism inside a group. When the customers

arrive at the MSCCC queue, they are queued in the order of

their arrival. Let k be a group index. A customer from group kwill be allowed to enter service at the MSCCC queue provided

that one or more of the B servers is available and that at most

n−1 other customers from group k are already in service at the

MSCCC queue. An MSCCC queue can model systems where

2013 IEEE 21st International Symposium on Modelling, Analysis & Simulation of Computer and Telecommunication Systems

1526-7539/13 $26.00 © 2013 IEEE

DOI 10.1109/MASCOTS.2013.21

131

customers simultaneously occupy two resources. The system

resources are partitioned into K primary and B secondary

resources. Even more complex disciplines may be represented

(see the reference at the end of chapter 2 in [1]).

OI queues are not symmetric in general but some of them

are (for instance, Infinite Server queues, M/M/K queues, PS

queues). They are not reversible but they are quasi-reversible

as stated in [1]. Thus they have a steady-state distribution with

product form. Note that in general, the closed form solution

contains a normalizing constant, which is hard to compute.

Some MVA like algorithm have been proposed to analyze

closed networks of MSCCC queues [5].

The theory of queues with signals has received a con-

siderable attention since the seminal paper of positive and

negative customers [6], published by Gelenbe 20 years ago.

Traditional queueing networks are used to represent contention

among customers for a set of resources (one resource for a

simple queue, several distinct resources for a MSCCC queue).

Customers move form server to server, they wait for service,

but they do not interact among themselves.

Signals are used to change these rules. In a network of

queues with signals (also denoted as a G-network of queues)

customers are allowed to change to signals at the completion

of their service and signals interact at their arrival into a queue

with customers already present in the queue. However signals

are never queued. They try to interact (they may fail) and

disappear immediately. Despite this deep modification of the

model, G-networks still preserve the product form property

for the steady-state distribution of some queueing networks

with Markov routing. It must be clear that the results are

more complex than Jackson networks. The G-networks flow

equations exhibit some uncommon properties: they are neither

linear as in closed queueing networks nor contracting as

in Jackson queueing networks. Therefore the existence of a

solution had to be proved [7] by new techniques from the

theory of fixed point equation. A numerical algorithm was

also developed [8].

The first type of signal [6] was introduced as a negative

customer. A negative customer deletes a positive customer at

its arrival at a backlogged queue. Many new signals associated

with networks with a product form solution have been studied

so far: triggers which redirect other customers among the

queues, catastrophes which flush all the customers out of a

queue [9], [10], resets [11], synchronized arrivals in a set of

queues [12], signals which change the class of the customer

in service [13] or which change the phase of the customer

in service for Phase type service distribution. Multiple class

versions of some models have also been derived [14], [15].

G-networks had also motivated many new important results

in the theory of queues. As negative customers lead to cus-

tomer deletions, the original description of quasi-reversibility

by arrivals and departures does not hold anymore and new

versions have been proposed. At the time being, the description

proposed by Chao and his co-authors in [16] looks suffi-

cient to study queues with customers and signals. Another

approach, based on Stochastic Process Algebra, was proposed

by Harrison [17], [18]. The main results (CAT and RCAT

theorems and their extensions [19], [17], [18]) give some

sufficient conditions for product form stationary distributions.

This technique clearly has a different range of applications as

it allows to represent component based models which are much

more general and more detailed than networks of queues. Here

we use the quasi-reversibility characterization by Chao [16]

because it looks easier in the context of an abstract queueing

discipline.

The technical part of the paper is as follows. In section II,

for the sake of completeness, we first give a short introduction

to quasi-reversibility of queues with signals. Then in Section

III, we present the Order Independent set of disciplines (with-

out and with loss) and we restrict this set to allow negative cus-

tomers. Then, we consider the interconnection of such queues

in an open network and we study stochastic knapsack model

with negative customers. In Section IV, we present a more

complex type of signal denoted as negative signal. Negative

signals have the ability to delete several customers in several

queues. They can also trigger a customer movement between

queues. We prove that under some technical constraints on the

rates, OI queue are quasi-reversible. Thus, we state that such

an open network of queues has a product form steady-state

solution. Section V is devoted to some examples.

II. A BRIEF INTRODUCTION TO QUASI-REVERSIBILITY OF

QUEUES WITH SIGNALS

We first present a brief introduction (inspired from [12])

to quasi-reversible queues and how to interconnect them to

obtain networks with product form solution.

A. Definition of quasi-reversibility

In [16], chap 3, Chao, Miyazawa and Pinedo gave a defi-

nition of quasi-reversibility for queue with or without trigger.

Let us introduce the definition with trigger which includes

simultaneous events.

Consider a queue where the queue-content evolves as a

continuous time Markov chain on state space S . For a pair

of states (�x, �y), we decompose the transition rate function

q(�x, �y) of the queue into three types of rates: qAu (�x, �y), u ∈T ; qDu (�x, �y), u ∈ T ; qI(�x, �y), where T is the set of the classes

of arrivals and departures, which is countable. The transition

rate of the queue can be written as:

q(�x, �y) =∑u∈T

qAu (�x, �y) +∑u∈T

qDu (�x, �y) + qI(�x, �y), �x, �y ∈ S.

The transition rate functions qAu , qDu and qI generate the

point processes corresponding to class u arrivals, class udepartures and the internal transitions, respectively. “A”, “D”

and “I” stand for “arrival”, “departure” and “internal”.

Suppose that q admits a stationary distribution π. Further-

more, assume that when a class u arrives and changes the state

of the queue from �x to �y, it instantaneously triggers a class vdeparture with probability fu,v(�x, �y), where:

∑v

fu,v(�x, �y) ≤ 1, u ∈ T, �x, �y ∈ S.

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With probability 1 −∑v fu,v(�x, �y), the class u arrival does

not trigger any departure. Function fu,v(�x, �y) is the triggeringprobability. When

∑v∈T fu,v(�x, �y) ≡ 0 for all u ∈ T, �x, �y ∈

S , no instantaneous movement between queues may occur due

to signal.

The quasi-reversibility of queues with instantaneous move-

ment is defined as follows.

Definition II.1. If there exist two sets of non-negative numbers{αu, u ∈ T} and {βu, u ∈ T} such that: for all �x ∈ S , u ∈ T,

∑�y∈S

qAu (�x, �y) = αu, (1)

∑�y∈S

π(�y)[qDu (�y, �x) +

∑v∈T

qAv (�y, �x)fv,u(�y, �x)]

= βuπ(�x),

(2)

then the queue with signal is said to be quasi-reversible with

respect to {qAu , fu,v, u, v ∈ T}, {qDu , u ∈ T} and {qI} .

The non-negative numbers αu and βu are called the arrival

rate and departure rate of class u customers. Chao et al. proved

that this definition of queue without instantaneous movements

is equivalent to the quasi-reversible definition given by Kelly in

[20]. This implies that the arrival processes and the departure

(triggered and non-triggered) of class u customers are Poisson

processes.

B. Network of quasi-reversible queues with signals and in-stantaneous movement

Consider a network of N queues. Each of them is a quasi-

reversible queue with signals as described previously. The set

of arrival and departure classes is T .

Let �xi be the state of queue i. Let Si be the state space. The

Poisson source has index 0 and we assume that the source has

only one state which is denoted as 0. For each queue, we need

to specify the arrival effects, the departure rate, the internal

transition rate and the triggering probability. For queue i, we

introduce functions pAiu, qDiu, qIi and fiu,v on the state space

Si:• pAiu(�xi, �yi) is the probability that a class u arrival at queue

i changes the state from �xi to �yi, where it is assumed that∑�y∈Si

pAiu(�xi, �yi) = 1, �xi ∈ Si;• qDiu(�xi, �yi) is the rate at which class u departures change

the state of queue i from �xi to �yi;• qIi (�xi, �yi) is the rate at which internal transitions change

the state of queue i from �xi to �yi;• fiu,v(�xi, �yi) is the triggering probability that when a class

u arrival occurs at queue i and the state changes from �xi

to �yi, it simultaneously induces a class v departure, where∑v∈T fiu,v(�xi, �yi) ≤ 1, i ≤ N, u ∈ T, �xi, �yi ∈ Si.

For source 0, we set pA0u(0, 0) = 1, pA0u(0, 0) = β0u,

qI0(0, 0) = 0 and f0u,v ≡ 0. Here, βA0u is the arrival rate

to the network from the outside (the source).

In Chao’s model, a queue is defined by three rates qAu , qDuand qI . In that case, the arrival effect function may be defined

as:

pAu (�x, �y) =qAu (�x, �y)∑z q

Au (�x, �z)

,

and qDu and qI are the departure and internal transition

functions.The dynamics of the network are described as follows.

Customers of class u arrive to the network from the outside

(the source) according to a Poisson process with rate β0u, and

are routed to queue i as a class v arrival with probability r0u,iv .

A class u departure from queue i, either trigger or non-trigger,

enters queues j as a class v arrival with probability riu,jv . It

is assumed that:N∑j=0

∑v

riu,jv = 1, i = 0, 1, . . . N, u ∈ T.

Furthermore, whenever there is a class u arrival at queue

i, either from the outside or from other queues, it makes

the state of the queue change from �xi to �yi with probability

pAiu(�xi, �yi), it also triggers a class u departure with probability

fiu,v(�xi, �yi), and it triggers no departure from queue i with

probability 1−∑v∈T fiu,v(�xi, �yi), i = 0, 1, . . . , N.

The transition rate function of the network is denoted by

q(�x, �y), �x, �y ∈ S = S1 × · · · × SN (note that we accept the

case where q(�x, �x) �= 0).

Consider for each queue i the following auxiliary process:

q(�αi)i (�xi, �yi) =

∑u∈T

(αiup

Aiu(�xi, �yi)+qDiu(�xi, �yi)

)+qDi (�xi, �yi),

where (�αi) = (αiu, u ∈ T ) are considered as dummy

parameters. Suppose that q(�αi)i has a stationary distribution

π(�αi)i . We always have:

∑�yi∈Si

αiupAiu(�xi, �yi) = αiu, i = 1, . . . N, u ∈ T.

Hence, the quasi-reversibility of q(�αi)i for i = 1, . . . , N is

equivalent to the existence of a set of non-negative numbers

βiu, u ∈ T such that:∑�yi

π(�αi)i (�yi)

[qDiu(�yi, �xi) +

∑v∈T

αivpAiv(�yi, �xi)fiv,u(�yi, �xi)

]

= βiuπ(�αi)i (�xi), (3)

for all �xi ∈ Si, i = 1, . . . , N and u ∈ T .Queue i in isolation is said to be quasi-reversible with �αi

if (3) is satisfied.Since αiu and βiu are the arrival and the departure rates

of class u customers at queue i, we have the following traffic

equations:

αiu =N∑j=0

∑v

βjvrjv,iu , i = 0, 1, . . . , N. (4)

We need the following condition to ensure that the network

process is regular:

N∑i=1

∑xi∈Si

π(�αi)i

∑�yi∈Si

q(�αi)i (�xi, �yi) <∞.

133

The stationary distribution of the network process has

product form (see [16] for a proof).

Theorem II.2. Under the assumptions on the routing, if eachqueue with signals is quasi-reversible (with �αi as the solutionto the traffic equations (4)), then the queueing network withsignal has the product form stationary distribution

π(�x) =N∏i=1

π(�αi)i (�xi),

where π(�αi)i is the stationary distribution of q

(�αi)i , i =

1, . . . , N .

III. NETWORK OF OI QUEUES WITH NEGATIVE

CUSTOMERS

We formally present an Order Independent queue following

[1] and we introduce a constraint to model an OI queue with

negative customers. Then, we prove that such a queue is quasi-

reversible. Thus, open networks of restricted OI queues with

negative customers have a product form stationary distribution

when they are stable. Finally, we present some examples.

A. An OI queue

We consider a finite set C for the classes of customers.

Customers of type c arrive according to a Poisson process

of rate λc. Customers of type c ask for an exponential service

with mean 1/μc. For simplification, let us denote by λ the

sum of all λc: λ =∑

c∈C λc. The state space is given by

S = {∅} ∪ {(cn, · · · , c1) | ci ∈ C}.Consider a queue in state (cn, · · · , c1). The total ser-

vice effort will be supplied at rate φ(cn, · · · , c1). A portion

γi(cn, · · · , c1) of the total service effort is directed at the cus-

tomer in position i. A customer arriving at a queue with size nwill be inserted at position n+1. When a customer in position

i completes its service and leaves, the customers in position

i+1, · · · , n move to position i, · · · , n− 1, respectively. Note

that∑

i γi(cn, · · · , c1) may be strictly smaller than 1, thus

one can model a waste of the power of the server.

The departure rate of the customer in the queue in po-

sition i is given by φ(cn, · · · , c1)μciγi(cn, · · · , c1). For all

(cn, · · · , c1) ∈ S and for all 1 ≤ i ≤ n, the rates of service

completion can be rewritten as:

φ(cn, · · · , c1)μciγi(cn, · · · , c1) = μ(n)si(cn, · · · , c1),such that:

1) si(cn, · · · , c1) = si(ci, · · · , c1), for any 1 ≤ i ≤ n,

which means that the service rate of customer at position

i only depends on the first i entries of the state vector.

2) k(cn, · · · , c1) =∑n

1 si(ci, · · · , c1) is independent of

the exact ordering of (cn, · · · , c1), which means that

the service rate is the same if we change the ordering

of the customers in the queue.

3) μ(n) > 0, for n > 0, and s1(c) > 0 for any c ∈ C. The

service rate is always positive for a backlogged queue.

This is the usual definition of an OI queue. One can easily

check that FCFS (with one class of customer), Processor

Sharing, Infinite Server queues are Order Independent queues.

For more complex discipline like MSCCC and MHSCC, see

[1] or the following of this paper.

We will also need the following property. Its proof is

omitted as it has already been published in Chapter 2 of [1]:

Property III.1. We have by induction on n:

k(cn, . . . , c1) = sn(cn, . . . , c1) + k(cn−1, . . . , c1).

Now we will restrict to the queueing disciplines which are

consistent with negative customers.

Definition III.2. We say that an OI queue is consistent withnegative customers if k(cn, · · · , c1) is upper bounded by Bfor all (cn, · · · , c1).

Such a constraint precludes to consider Infinite Server

queues. Indeed for an IS queue, k(cn, · · · , c1) linearly in-

creases with the population in the queue. Therefore it is

not upper bounded and we cannot represent IS queues with

negative customers. In this section all the OI queues are

supposed to be consistent with negative customers.

Let us now describe the queue we consider. First, we assume

that the queue is order independent. Negative customers arrive

according to a Poisson process of rate λ−. A negative customer

deletes a positive customer when there is any. The selection

of the customer to be deleted mimics the repartition of the

power of the server. More precisely, when the state of the

queue is (cn, · · · , c1), a customer in position i will be selected

for cancellation with probability:

si(cn, · · · , c1)B

.

Note that as B is an arbitrary upper bound of

k(cn, · · · , c1) =∑n

1 si(ci, · · · , c1), this allows that the se-

lection of customers to be deleted may fail with a positive

probability which is state dependent and equal to

(B −∑n1 si(ci, · · · , c1))

B.

We now prove that the queue is quasi-reversible under stability

constraints.

Theorem III.3. Consider the measure on S:

π(cn, · · · , c1) = π(0)

n∏i=1

λci

(μ(i) + λ−/B)k(cn, · · · , c1) , (5)

where π(0) is a positive real number. The OI queue withnegative customer is stable if and only if

G =∑

(cn,··· ,c1)∈S

n∏i=1

λci

(μ(i) + λ−/B)k(cn, · · · , c1) <∞.

Moreover the stationary distribution is given by (5) withπ(0) = 1/G and the queue is quasi-reversible.

134

Proof: First, we prove that the measure defined in (5)

is an invariant measure, (i.e. the global balance equations are

satisfied) which means that:

π(0)∑c∈C

λc =∑c∈C

π(c)[μ(1)k(c) + λ−k(c)/B],

and for a non empty state (cn, · · · , c1):π(cn, · · · , c1) [

∑c∈C λc + (μ(n) + λ−/B)k(cn, · · · , c1)]

= λcnπ(cn−1, · · · , c1)

+∑

c∈C∑n

i=0 π(cn, · · · , ci+1, c, ci, · · · , c1)

×[μ(n+ 1) + λ−/B]

×si+1(cn, · · · , ci+1, c, ci, · · · , c1).The proof is very similar to the proof in case of OI queue

without negative customers which is published in [1]. For the

sake of completeness and readability we give the proof in the

appendix. Remark that quasi-reversibility is proved as:

n∑i=0

π(cn, · · · , ci+1, c, ci, · · · , c1)π(cn, · · · , c1) (μ(n+ 1) +

λ−

B)

×si+1(c, ci, · · · , c1) = λc.

B. OI queue with loss

In this section, we consider OI queue with loss. Note that

this does not imply that the queue capacity is finite. Indeed,

due to contention between customers, some types of customer

may enter the queue while some others are not allowed to

do. Thus, we need a more abstract specification of the states

and we still have to study the stability. Of course, finite

capacity queues may still be represented by a loss queue. In

the following, the state space is denoted by S̃, which is a

subset of S. We assume that the following conditions holds:

• if (cn, · · · , c1) ∈ S̃, then (cσ(n), · · · , cσ(1)) ∈ S̃ for any

permutation (σ(n), . . . , σ(1)) of n, · · · , 1 ,

• if (cn, · · · , c1) ∈ S̃, then (cn−1, · · · , c1) ∈ S̃ .

We further assume that all the properties for OI infinite queue

are still satisfied for OI queue with loss using state space S̃.

We also have to describe the arrival of customers when the

queue is rejecting customers. Let us first define a blocking

state.

Definition III.4. A state (cn, · · · , c1) is blocking for classc, if and only if customers of class c are rejected when theyarrive to the queue at state (cn, · · · , c1). A state (cn, · · · , c1)is blocking if and only if there exists a customer class (say c),such that the state is blocking for class c.

It is now possible to define the behavior of the queue when

it is in a blocking state. We assume that a customer of class

c, arriving to a queue in a state blocking for class c, jumps

immediately over the queue and continues the routing as a

customer of class c at the completion of its service.

Theorem III.5. The OI queue with loss and negative customeris stable if and only if

G = 1 +∑

(cn,··· ,c1)∈S̃

n∏i=1

λci

(μ(i) + λ−/B)k(cn, · · · , c1) <∞.

Note that when the state space S̃ is finite, this condition clearlyholds. But we can also represent a loss queue which an infinitestate space. The stationary distribution is given by (5)

π(cn, · · · , c1) = 1

G

n∏i=1

λci

(μ(i) + λ−/B)k(cn, · · · , c1) , (6)

and the queue is quasi-reversible.

Proof:

1) If at state (cn, · · · , c1), one can accept an arrival of a

class c-customer, we have, as in Theorem III.3:

n∑i=0


λ−

B)

×si+1(c, ci, · · · , c1) = λc.

2) If state (cn, · · · , c1) is a blocking state for class c-customers, we clearly get:

π(cn, · · · , c1)π(cn, · · · , c1)λc = λc.

Thus the queue is quasi-reversible.

C. Stochastic knapsack model

The stochastic knapsack model was introduced by Ross

[21]. It is an example of OI queue with loss. There are

K resources to share. The arrival rate and service rate for

customer of class c are given by λc and μc. There are at most

Bc customers of class c present in the queue. Customer of

class c holds bc resource units. If there are Bc customers of

class c, then the arriving customer of class c will be rejected. If

the number of units in use is larger than K− bc, then arriving

customer of class c will be rejected too.

Using the result for OI queue with loss, one has that:

π(cn, · · · , c1) = π(0)n∏

i=1

λci∑ij=1 μcj

.

However, remark that in this case, the order of the customer

has no importance. This is why Ross modelled the queue with

the numbers of customers for each class (say (mc)c∈C).

We can prove by induction that the sum of

π(cσ(n), · · · , cσ(1)) over all permutation σ of (1, · · · , n) is:

∑(σ(1),··· ,σ(n))

π(cσ(n), · · · , cσ(1)) = π(0)∏ (λc)

mc

(μc)mc(mc)!,

which is exactly the result proved by Ross.

We can also consider the knapsack model with negative

customer. Negative customer arrives according to a Poisson

process of rate λ−. A customer of class c will be chosen as a

135

target with probability mcλ−/K. The stationary distribution

is given by:

π((mc)c∈C) = π(0)∏ (λc)

mc

(μc + λ−/K)mc(mc)!.

D. Interconnection of OI queues with negative customersWe proved that the model of a single queue is quasi-

reversible with respect to each type c for departures caused

by both a service completion and a cancellation. Hence, we

can consider the network model and we can apply Theorem

II.2 for network of quasi-reversible queues.Consider an open and connected network of N OI queues,

with or without loss. We only consider open topologies be-

cause in a closed topology, the number of positive customers

cannot increase with fresh arrivals, while negative customers

will delete positive customers. Therefore the limit for a

strongly connected closed network is an empty network.In queue l, the arrival rate and service rate for customer of

class c are given by λlc and μl

c. The arrival rate for negative

signals is λl,−. The total service effort in queue l is given by

function φl and the proportion function is γl. The function kl

is supposed upper bounded by Bl. The state space of queue lis given by S̃ l (or Sl when we consider a queue without loss).

A customer of class c in queue l after service completion

can move to another queue l′ as a customer of class c′ with

probability P l,l′c,c′ or as a negative customer with probability

P l,l′c,−. It can also leave the network with probability dlc. We

have the following condition:∑l′ �=l

(P l,l′c,− +

∑c′∈C

P l,l′c,c′) + dlc = 1. (7)

We can apply Theorem II.2 on networks of quasi-reversible

queues to get the following result. Again we omit the proof,

which is a simple application of this theorem.

Theorem III.6. If we have

Gl =∑

(cn,··· ,c1)∈S̃l

n∏i=1

Λlc

(μl(i) + Λl,−/Bl)kl(ci, · · · , c1) <∞,

then the stationary distribution of the network is given by:for all state �c = (�c 1, · · · ,�c N ) where �c l = (clnl

, · · · , cl1)

π(�c) =

N∏l=1

πl(�c l), (8)

where:

πl(�c l) = πl(0)nl∏i=1

Λlc

(μl(i) + Λl,−/Bl)kl(cli, · · · , cl1)., (9)

where Λlc and Λl,− are solutions of the network Traffic

Equations:

Λlc = λk

c +∑l′ �=l

∑c′∈C

Λl′c′P

l′,lc′,c, (10)

Λl,− = λl,− +∑l′ �=l

∑c∈C

Λl′c P

l′,lc,−. (11)

IV. PROPAGATION OF NEGATIVE SIGNALS IN A NETWORK

OF OI QUEUES

In this section, we consider OI queues with negative signals.

Following Harrison [18] and Chao et al [16], we define

negative signals as follows: when a negative signal deletes a

customer at its arrival in a queue (if the queue is not empty),

then the deleted customer joins immediately the next queue

as a negative signal or a (positive) customer according to the

routing probability matrices defined in the following. If the

queue is empty, the negative signal vanishes instantaneously.

Thus, a negative signal may delete several customers in

several queues. As it is possible for the negative signal to loop

at a queue, it may be possible that a negative signal deletes

all the customers in a queue. Such an event is denoted as a

catastrophe [15]. It is also possible like in [22] to add a queue

to model a service and a delay between each iteration of the

deletion process. We first have to make some restrictions on

the set of queueing disciplines we consider.

Definition IV.1. We say that an OI queue is consistent withnegative signals if k(cn, · · · , c1) is upper bounded by B forall (cn, · · · , c1) and μ(n) = μ.

Note that this definition is more restrictive that Definition

III.2 as we add μ(n) = μ for all n.

Remark IV.2. In the case of MSCCC queue [1], μ(n) is equal

to 1 and si(cn, · · · , c1) is supposed to be μ if the customer

in position i is in service. Thus, even with the restrictions of

Definition IV.1, it is still possible to model complex queues.

In a general OI queue with negative customer, we proved

in the previous section that we have the quasi-reversibility

property for the departure of a customer caused by a real

departure after service completion and a departure due to a

cancellation. Indeed, we have:n∑

i=0


λ−

B)

×si+1(c, ci, · · · , c1) = λc.

In this section, we consider only the case in which μ(n) is a

constant. Hence it implies that the quasi-reversibility holds for

each type of customer for a real departure or for a departure

by a cancellation.

For a real departure of customer of type c caused by a

service completion, we have:

n∑i=0

π(cn, · · · , ci+1, c, ci, · · · , c1)π(cn, · · · , c1) μ

×si+1(c, ci, · · · , c1) = λcμ

μ+ λ−/B,

and for a departure of type c customer due to a negative signal,

we obtain:n∑

i=0

π(cn, · · · , ci+1, c, ci, · · · , c1)π(cn, · · · , c1)

λ−

B

×si+1(c, ci, · · · , c1) = λcλ−/B

μ+ λ−/B.

136

In the case of OI queue consistent with negative signals,

we can consider a network with a more flexible routing. We

keep the same assumptions for the arrivals and service, except

that μl(n) = μl. We also consider negative signals, which can

propagate in the network. Negative signals arrive to queue laccording to a Poisson process of rate λl,−.

As before, customer of class c in queue l after service

completion can move to another queue (say l′) as a customer

of class c′ with probability P l,l′c,c′ or as a negative customer with

probability P l,l′c,−, or can leave the network with probability dlc.

We still have the condition of Eq. 7. Again, a negative signal,

which arrives to queue l in state (cn, · · · , c1), will chose a

customer in position i as a “target” with probability

sli(cn, · · · , c1)Bl

.

But now, if the cancellation is successful, then the negative

signal will trigger the cancelled customer to another queue

(say l′) as a customer of class c′ with probability P l,l′−,c′ or

as a negative signal with probability P l,l′−,−, or to the outside

of the network with probability d−,l. Of course we have the

following normalization condition on the probabilities:

∑l′(P l,l′−,− +

∑c′∈C

P l,l′−,c′) + d−,l = 1.

As mentioned, the model of a single queue with negative

signals is quasi-reversible with respect to each type (c) of

customer after service completion and with respect to each

type (c) of customer after a cancellation. Hence, we can apply

Theorem II.2. The network Traffic Equations are given by:

Λlc = λk

c +∑l′ �=l

∑c′∈C

Λl′c′

μl′

μl′ + Λ−,l′c′ /Bl′

P l′,lc′,c

+∑l′

∑c′∈C

Λl′c′

Λ−,l′/Bl′

μl′ + Λ−,l/Bl′ Pl′,l−,c, (12)

Λl,− = λl,− +∑l′ �=l

∑c′∈C

Λl′c′

μl′

μl′ + Λ−,l′c′ /Bl′

P l′,lc′,−

+∑l′

∑c′∈C

Λl′c′

Λ−,l′/Bl′

μl′ + Λ−,l/Bl′ Pl′,l−,−. (13)

And the quantities Λlc and Λl,− appear in the product form

theorem.

Theorem IV.3. Consider Gl previously defined. If

Gl =∑

(cn,··· ,c1)∈S

n∏i=1

Λlc

(μl + Λl,−/Bl)kl(ci, · · · , c1) <∞

then the stationary distribution of the network is given by:for all state �c = (�c 1, · · · ,�c N ), �c l = (clnl

, · · · , cl1)

π(�c) =

N∏l=1

πl(�c l) (14)

where


Λlc

(μl + Λl,−/Bl)kl(cli, · · · , cl1). (15)

V. SOME EXAMPLES OF NETWORK OF OI QUEUES WITH

SIGNALS

A. A model for load balancing with cancellation of jobs

We consider a system with K inhomogeneous servers. The

service rates are respectively μ1, μ2, . . ., μK . We assume

without loss of generality that μ1 ≥ μ2 ≥ μi ≥ μK . All the

customers belong to the same class. The arrival rate is λ, and

we assume that customers ask for an exponential service with

mean 1. An arriving customer joins the end of the queue, it

begins its service with the server with the higher service rate,

if there is any available server. When a customer served by

server i completes its service and leaves, the following load

balancing mechanism takes place:

• If there is at least one customer waiting for service in the

queue, it joins server i.• If there is not such a customer, the customer being served

by the slowest server moves to server i (if server i is not

the slowest one).

More formally we get:

• si(cn, · · · , c1) = si(ci, · · · , c1) = μi, for any 1 ≤i ≤ min(n,K). Clearly, the service rate of customer at

position i in queue only depends on i.• k(cn, · · · , c1) =

∑min(n,K)1 μi is independent of the

exact ordering of the customers.

• μ(n) = 1 for n > 0.

Such a queue is obviously an OI queue and it represents a

queue with an ideal load balancing between heterogeneous

servers. As an OI queue, we know it has a product form

solution when it is combined into a network with Markov

routing with some other BCMP or OI queues or in general

quasi-reversible queues.

However, as we have only one type of customers, the state

of the queue can be written by the number of customers in

the queue: n. In the following, the only use the number of

customers in the queue for the state.

Now assume that this receives a flow of negative customers

coming from outside or from other queues. Let λ− be the

rate of arrival for these negative customers. First we can see

that the queue is consistent with negative customers. We take

B =∑K

i=1 μi. The negative customers act as described in

section II but we also have to take into account the policy

used to manage the queue: a customer moves instantaneously

to the fastest server available. More precisely, with probability

μi/B a negative signal interacts with server i. If it is empty,

the signal disappears immediately without any action. If server

i contains a customer, the signal deletes it. But the deleted

customer is replaced by a backlogged customer if there is

any or by the customer served by the slowest server. Finally,

one can get another interpretation of the combination of these

immediate transitions at state n:

137

• If n ≤ K, the negative customer deletes the slowest

customer with probability (∑i

k=1 μk)/B or disappears

without any effect with the remaining probability. This is

consistent with the usual rule on the empty queues which

are not impacted by negative customers.

• Otherwise, the negative customer deletes a backlogged

customer with probability 1.

.

The stationary of the model with negative customers is given

by:

π(n) = π(0)

n∏i=1

λ

(1 + λ−/B)∑min(i,K)

k=1 μk

Thus, this queue represents an ideal load balancing system

where cancellation of jobs can occur. These cancellations

are modeled by negative customers. The cancellation policy

described previously allows to delete backlogged customer or

customers served by the slowest server. Thus, the OI queue,

even if the description of the deleted customers is not the

same, gives the same Markov chain and the same steady-state

distribution.

Finally, note that this queue also has nice properties to help

the comparison of stochastic processes. Let us first define the

stochastic monotonicity (for more results, see [23]).

Definition V.1 (Stochastic ordering on a discrete state space).Let p and q be two distributions, p ≥st q if and only if∑

j≥k pj ≥∑

j≥k qj for all k.

Definition V.2 (Stochastic Monotonicity of a Markov Process

on a discrete state space). A Markov process {X(t)} on N ismonotone if X(0) = 0 then X(t) ≥st X(s) for all t > s > 0.

Proposition V.3. The queue with K inhomogeneous server,Exponential services, Poisson arrivals of customers and Pois-son arrivals of negative customers is stochastically monotone.

Proof: Indeed this queue is modeled by a birth and death

process and all these processes are proved to be stochastically

monotone [24].

As a stochastically monotone Markov process, it allows

to compare this ideal load balancing queue with a real load

balancing mechanism. Such an idea will be explore in a sequel

of that paper.

B. A model for concurrent access with synchronized cancel-lations of requests

More precisely, in the language of queueing networks we

consider networks of MSCCC queues and MSHCC queues

with negative customers. As mentioned in the introduction,

the Multiserver Station with Concurrent Classes of Customers

(MSCCC) queue is an example of OI queue. At the completion

of a service a customer is sent as a negative customer to

another queue.

Let us begin with the MSCCC queues. At queue l, the

arrival rate and service rate for customers of type c are λlc

and μl. Customers are queued for service in the order of their

arrival. The concurrency constraint is the following: at most

Kl customers can be in service and at most Blc customers of

type c can be in service. The arrival rate for negative customers

in queue l is λl,−. As first we have to check that a MSCCC

queue is consistent with negative customers.

In this case, μl(n) = 1, and

sli(cn, · · · , c1) = μl1li(cn, · · · , c1),

where 1li is the indicator function, which is 1 if in queue l

at state (cn, · · · , c1), the customer in position i is in service,

and 0 otherwise. Then

kl(cn, · · · , c1) = μl(Kl ∧∑c∈C

(M lc ∧Bl

c)),

where M lc is the number of customers of type c in queue l

and a ∧ b = min(a, b).We have clearly that kl is upper bounded by μlKl. Thus, we

will use the bound Bl = μlKl. Note that the bound is tight.

This bound shows us that the MSCCC queue is consistent

with negative customers. Let us now consider a network of NMSCCC queues. The Traffic Equations are still described by

Equation (10,11) and the stationary distribution of the network

will be given by:

π(�c) =N∏l=1

πl(�c l), (16)

where


Λlc

(μl + Λl,−/Kl)(Kl ∧∑c∈C(M l

c ∧Blc)

.

(17)

To obtain a MHSCC queue, we have to further decompose

the classes of customers. But this decomposition into several

sub-classes must only be done in a hierarchical way. Class cis decomposed into sub-classes c(1), . . . c(q). Each level of

the decomposition allows to add more capacity constraint.

Thus we may require that the number of class c customers in

service is smaller than Bc while the number of sub-class c(i)customers in service is smaller than Bc(i). Of course we have

Bc ≤∑

Bc(i) to obtain a more complex queueing mechanism.

This is denoted as a MSHCC (Multi Server Hierarchical

Concurrent Customer) queue in [1].

The distribution can be given while we take into account

that the service rate in this case is given by:

kl(cn, · · · , c1) = μl(Kl ∧∑c∈C

∑c(∗)

(mlc(∗) ∧Bl

c(∗))),

where mlc(∗) is the number of customers of sub-class c(∗) in

queue l.We slightly change the presentation. The global set of

classes is then represented as a tree. The leaves of the tree

are the classes of customers (C̃) plus an integer (B∗̃) which

describes number of customers of this class which can be in

service. An internal node of the tree represent a set of classes.

The arcs between nodes describes the inclusion of subsets into

138

2

2

2

3

2

5

4

7

Class1

Class 2

Class 3

Class 4

Class 5

Fig. 1. A tree of constraints among classes of customers in a MSHCC queue.

a set. An internal node contains the upper bound on the number

of customers of this set which can be in service (see Fig. 1).

The discipline represented by this tree is the following: the

number of servers is 7 (the value of the root node of the

tree), at most two customers of each class can be in service

at the same time, except class 5 whose upper bound is 3.

Finally the cumulated number of classes 1 to 3 customers in

service must be smaller than 5 and the cumulated number of

customers of class 4 and 5 is smaller than 4. Clearly we can

represent many constraints between customers with this type

of hierarchical structure. When this graph is a tree, we obtain

a MSHCC queue. The service rate in this case is given by:

k(c̃n, · · · , c̃1) = μ{K ∧(

5 ∧ (m(1) ∧ 2 +m(2) ∧ 2 +m(3) ∧ 2)

+4 ∧ (m(4) ∧ 2 +m(5) ∧ 3))}

,

where m(c̃) is the number of customers of class c̃.The distribution is given by:

π(c̃n, · · · , c̃1) = π(0)

n∏i=1

λc̃

(μ+ λ−/K)k(c̃n, · · · , c̃1) .

VI. CONCLUDING REMARKS

Networks with signals have been used in several application

domain: random neural networks with an associate efficient

learning algorithm, texture generation, optimization, subjective

quality of service for video and audio transmission, chemical

reaction modeling, gene network modeling and of course the

cognitive packet technology for network routing (for a list of

references, see for instance [25]). Most of these applications

are based on the theory of networks with positive and negative

customers, i.e. the simplest interaction between customers and

signals. The results we have proved here, are the first ones,

to the best of our knowledge, which also allow complex

interactions between customers (for instance the blocking in

MSCCC queues and MSHCC queues) and between customers

and signals. They also have some deep relations with the

model of cooperation of Markov chains [26] or Markov chains

in competition [27] We hope that all these results will open

avenues for new applications of G-networks.

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[16] Chao, X., Miyazawa, M., Pinedo, M.: Queueing Networks Customers,Signals and Product Form solutions. John Wiley & Sons (1999)

[17] Harrison., P.: Turning back time in Markovian process algebra. Theo-retical Computer Science 290(3) (2003) 1947–1986

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[20] Kelly., F.P.: Reversibility and Stochastic Networks. J. Wiley (1987)[21] Ross, K.W.: Multiservice loss models for broadband telecommunication

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Appendix

We now give the proof of Theorem III.3. We need to prove

the global balance equations given in section III.

The queue is quasi-reversible if we can find a collection of

numbers π(cn, · · · , c1) which are all positive, which sum to

unity and which satisfy the global balance equations, such that

for all state (cn, · · · , c1) and for all class c:∑n

i=0π(cn,··· ,ci+1,c,ci,··· ,c1)

π(cn,··· ,c1)

×(μ(n+ 1) + λ−/B)

×si+1(c, ci, · · · , c1) = βc

(18)

In fact, we will prove that βc = λc. Combining Eq. 18 with

βc = λc and with the global balance equation we obtain:

(μ(n) + λ−/B)k(cn, · · · , c1)π(cn, · · · , c1) =λcnπ(cn−1, · · · , c1) . (19)

The solution given in (5) is clearly a solution to this

equation. We will now prove that this solution is also a solution

to:n∑

i=0


λ−

B)

×si+1(c, ci, · · · , c1) = λc. (20)

We prove by induction on n that equality (20) holds for all

(cn, · · · , c1) ∈ S and c ∈ C.

Consider first the empty queue (n = 0). We substitute the

solution proposed in Eq. (5) to the right-hand side of (20), and

noting that k(c) = s1(c), it yields:

μ(1)π(c)s1(c)

π(0)=

μ(1)π(0)s1(c)

λcπ(0)μ(1)k(c)= λc,

so that the base of the induction is proved.

Next assume that the solution given by Eq. (5) satisfies Eq.

(20) up to value n − 1 of the summation index and consider

this relation (i.e. Eq. (20)) for the index equal to n. The left-

hand side of (20) (denoted in the following lhs for the sake

of readability) can be written as:

lhs =

n∑i=0


λ−

B)

×si+1(c, ci, · · · , c1)

=n−1∑i=0


λ−

B)

×si+1(c, ci, · · · , c1)+π(c, cn, · · · , c1)π(cn, · · · , c1) (μ(n+ 1) +

λ−

B)

×si+1(c, ci, · · · , c1).We substitute the value of π given by Eq. (5) to obtain:

lhs =(μ(n) + λ−/B)k(cn, · · · , c1)

μ(n+ 1) + λ−/B

×n−1∑i=0

π(cn−1, · · · , ci+1, c, ci, · · · , c1)π(cn−1, · · · , c1)k(cn, · · · , ci+1, c, ci, · · · , c1)

×(μ(n+ 1) + λ−/B)si+1(c, ci, · · · , c1)+

λc

k(c, cn, · · · , c1)sn+1(c, cn, · · · , c1)

Now remember that we consider an OI queue. Therefore

we have that:

k(cn, · · · , ci+1, c, ci, · · · , c1) = k(c, cn, · · · , c1).Furthermore k(c, cn, · · · , c1)(μ(n+ 1) + λ−/B) > 0. Thus,

lhs =k(cn, · · · , c1)k(c, cn, · · · , c1)

n−1∑i=0

π(cn−1, · · · , ci+1, c, ci, · · · , c1)π(cn−1, · · · , c1)

×(μ(n) + λ−/B)si+1(c, ci, · · · , c1)+λcsn+1(c, cn, · · · , c1)

k(c, cn, · · · , c1) .

Now we apply the induction assumption on the right-hand

side of the above equation and we substitute λc, which yields

after factorisation:

lhs = λck(cn, · · · , c1) + sn+1(c, cn, · · · , c1)

k(c, cn, · · · , c1) .

Thus, by Remark III.1 we obtain lhs = λc, which completes

the induction.

Thus, equations (19) and (20) are satisfied. The balance

equation is proved by taking the sum. Equations 20 also

proves the quasi-reversibility of the queue with respect to

the departure of customers of type c caused by a service

completion or a cancellation due to a negative customer.

This completes the proof.

140

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