Equilibrium Strategies for Overtaking-Free Queueing Networks under Partial Information

Abstract

We investigate the equilibrium strategies for customers arriving at overtaking-free queueing networks and receiving partial information about the system’s state. In an overtaking-free network, customers cannot be overtaken by others arriving after them. We assume that customer arrivals follow a Poisson process and that service times at any queue are independent and exponentially distributed. Upon arrival, the received partial information is the total number of customers already in the network; however, the distribution of these among the queues is left unknown. Adding rewards for being served and costs for waiting, we analyze the economic behavior of this system, looking for equilibrium threshold strategies. The overtaking-free characteristic allows for coupling of its dynamics with those of corresponding closed Jackson networks, for which an algorithm to compute the expected sojourn times is known. We exploit this feature to compute the profit function and prove the existence of equilibrium threshold strategies. We also illustrate the results by analyzing and comparing two simple network structures.

1. Introduction

The analysis of optimal strategies for queueing systems can be traced back to 1969, when Naor found the optimal strategy for an individual that gives a rule for join or balk in the observable $M/M/1$ model [1]. This work was later extended by Edelson and Hildebrand in 1975, by looking at the unobservable $M/M/1$ model [2]. The economic analysis of queueing systems has since expanded to more general systems, such as the $M/G/1$ queue [3,4], focusing on systems with priorities and heterogeneous customers [5,6,7] or with multiple servers [8,9,10]. A nice introduction on queueing systems from the economic perspective and a recollection of a variety of results can be found in the recent monograph by Hassin and Haviv [11].

Most results in the literature focus on single-station systems. Except for a few cases, there is little research on an equilibrium strategy for multi-hop queueing networks. This is surprising from an application standpoint, as many real-world systems consist of multiple stages, each involving queueing and correlated waiting times. Examples include medical services requiring several tests, traffic systems with vehicles choosing different routes, or communication networks where data pass through multiple servers. The limited research stems from the mathematical complexity involved in the economic analysis of queueing networks.

The first attempt was made by D’Auria and Kanta [12], who made an economic analysis of a two-node Jackson tandem network. They assumed that the arriving customers received some information about the current state of the network, and using this information, they decided whether to join or balk the system. They considered the fully observable case, the unobservable one, and the partially observable case, in which the incoming customers only knew the total number of customers in the system but not how they were located in the nodes. For all cases, they showed that a pure threshold equilibrium policy exists and how to compute it. Later, B. Kim and J. Kim generalized this result by considering multi-hop tandem networks [13]. More recently, Refs. [14,15] analyzed extensions of the famous Lu–Kumar network [16] from an economic point of view by introducing route-choosing customers.

The focus of this work is to extend the results in [12,13] to a more general class of networks; in particular, we consider the class of overtaking-free Jackson networks by looking for equilibrium strategies in a partially observable case. That is, we assume that the arriving customers are informed only about the number of users already inside the network, and using this information, they decide whether to join or balk the system by evaluating a computed profit function. We show that this function exists and discuss when it is unique and of the threshold type. Moreover, we show that the profit function is an increasing function of the effective arrival rate, correcting an inaccuracy contained in [12]; see Appendix B.

Overtaking-free network, defined more in detail in Section 2, refers to the fact that no customer may be surpassed by anyone that joins the system at a later epoch. This property, fulfilled in any situation where the order of arrival is respected, is here mainly required for mathematical reasons as it allows for a very neat solution. Actually, removing this assumption is a very challenging problem, and we believe that it will not lead to a unique general solution but to the arising of a variety of situations depending on the network topology. The class of overtaking-free networks appeared in the queueing context already in 1979 when Walrand and Varaiya proved that in an open multiclass Jackson network, the sojourn times of a customer along an overtaking-free path are independent [17]. Kelly and Pollett obtained a similar result in 1983 analyzing a closed Jackson network [18]. The overtaking-free condition ensures that no customer can influence the sojourn time of any other customer who arrived earlier in the system. In this study, we characterize the topology of the FIFO overtaking-free networks with exponential service times, and show how to compute and optimize the profit function of partially informed arriving customers.

Another main feature of our model consists in revealing to customers incomplete information about the system. In the literature, these models are referred to as partially observable and their analysis has gained more attention since the 2000s. In particular, regarding single-service systems, this topic was discussed in 2007 by Guo and Zipkin, who looked at the effects on the model with no information, partial information, and full information [19]; and, in 2008, by Economou and Kanta, who explored the single-server Markovian queue with compartmentalised waiting space [20]. As for batch-service queueing systems, in 2017, Bountali and Economou studied the strategic customer behavior for the observable and unobservable $M/M/1$ queue with single arrivals and batch services [21], and, in 2019, they extended this model by considering the partially observable case, where the arriving customers observe only the number of complete batches that remain to be served [22].

The class of models analyzed by this work may find application in many fields, such as healthcare, communication systems, transportation, banking, and processing networks [14]. For example, in Section 6, we introduce model A (see Figure 2a), that may represent a healthcare center where clients enter the system, are first attended by a registration desk, and then, are scheduled to the correct doctor’s office. We also introduce model B (Figure 2b), that may represent a two-hop communication channel, where packets pass through two servers before reaching the destination.

Then, Section 2 contains the characterization of the overtaking-free networks, under the mentioned assumptions. In Section 3, we introduce the model and the notation. Then, in Section 4, we obtain the stationary distribution for the number of customers in the system in the unobservable and the observable cases and compute the expected sojourn time in any queue given the total number of customers in the system. In Section 5, we derive the equilibrium threshold strategies. In Section 6, we provide some numerical computations by comparing two different queueing networks. In Section 7, we discuss the limitations of our model and we highlight some possible directions for future research. Finally, in Section 8 we draw some conclusions.

2. Overtaking-Free Networks

In this section, we show that, under an FIFO discipline and with exponential service times, a necessary and sufficient condition for a queueing network to be overtaking-free is that its underlying graph is an out-tree.

We define by $G(V,E_0)$ the underlying finite directed graph of a queueing network. V represents the set of vertices, that correspond to the nodes where customers can queue up and receive their service. $E_0$ is the set of edges, that correspond to the allowed movements between queues and towards to the outside. We assume that $V\subset \mathbb{N}$ and $E_0\subset {\mathbb{N}}_0\times {\mathbb{N}}_0$, where ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$. We denote by $0\notin V$ the outside vertex, and we let $V_0=V\cup \left\{0\right\}$ and $E=E_0\cap (\mathbb{N}\times \mathbb{N})$. Sometimes, we refer to the graph $G(V_0,E_0)$ with the intention to include the outside node inside the set of vertices. $D\left(i\right)=\{j\in V_0:(i,j)\in E_0\}$ denotes the set of vertices in $V_0$ that are reachable from a vertex $i\in V_0$. After completing the service at queue $i\in V$, a customer either joins a queue among the nodes in $D\left(i\right)\cap V$ or leaves the system, whenever $0\in D\left(i\right)$. We let $d\left(i\right)=\left|D\right(i\left)\right|$ be the out-degree of vertex $i\in V_0$.

We assume that in the graph $G(V_0,E_0)$, the outside node can access and can be accessed by any node via a given path of edges. We also assume that the directed graph $G(V,E)$ is connected; that is, for any non-trivial subset $A\subset V$, there exists an edge $(i,j)\in E$ such that either $i\in A$ and $j\in V\backslash A$ or $i\in V\backslash A$ and $j\in A$. This implies that the network cannot be split into two separated subnetworks. If $d\left(0\right)=1$, we say that the graph $G(V,E_0)$ (as well as $G(V,E)$) is rooted. In this case, we assume that the root node is the vertex $1\in V$. The directed graph $G=(V,E)$, with root vertex $1\in V$, is an out-tree if $\left|E\right|=\left|V\right|-1$ and, for any $v\in V\backslash \left\{1\right\}$, there exists a directed path from 1 to v. It also holds that such a path is unique.

Now, we give a formal definition of an overtaking-free network [17,18]. We have in mind that the outside node 0 is a source of customers. We can think of it as being a single-server queue with an infinite number of customers in its line. In general, we are going to assume that this source of arriving customers is stochastically modeled as a Poisson process. We then define a tagged customer as a customer that just left the outside node 0 to reach any node in the set $D\left(0\right)\subset V$, that we are going to analyze.

Definition 1 (Overtaking-free network).

A queueing network is overtaking-free if and only if the sojourn time of a tagged customer in any queue along their path does not depend on the behavior of the customers arriving at the network at later times. That is, their sojourn time stays the same if no more customers are allowed to enter the network.

We define the following standing assumptions:

• The inter-arrival times, the service times, and the routing probabilities are independent random variables, whose distribution may only be node dependent;
• Each node is a single-server queue.

The above assumptions are implicitly assumed in the statement of the following theorem as well as for the rest of the paper.

Theorem 1 (Characterization of overtaking-free networks).

Under the FIFO discipline and exponential service times at each queue, a necessary and sufficient condition for a queueing network to be overtaking-free is that its underlying graph $G(V,E)$ is an out-tree.

Proof. 

It is clear that if the graph $G(V,E)$ is an out-tree and the service discipline is FIFO, then the network is overtaking-free. Indeed, each node would be reachable by a unique path, and along it, customers are served according to an FIFO discipline. On the other hand, assume that the network structure is not an out-tree, and consider two alternative paths joining two vertices $v_1,v_2\in V$. Then, there is a positive probability that two consecutive arriving customers go along the two alternative paths $0\to v_1\to v_2\to 0$, where the arrows denote a sub-path between two nodes. Since the support of the exponential distribution is $(0,\infty )$, there is a positive probability that the second customer, the one that arrived later, will reach node $v_2$ before the first customer, and will queue there in front of them. This would contradict the overtaking-free network hypothesis that the presence of the second user in the network should not alter the sojourn times of the first one. □

The previous proof would hold even if we substitute the exponential service distribution with any distribution whose support is unbounded or dense close to 0. However, in general, if we change the service discipline or the distribution of the service times, Theorem 1 may no longer be valid. One can easily construct counterexamples by using degenerate deterministic service times.

We conclude that under an FIFO discipline and with exponential service times in an overtaking-free network, node 1 serves as the root of the tree, and all other nodes have only one entering edge.

3. The Model

Under the assumption of exponentially distributed service times, an overtaking-free network is characterized by having a branching structure, described in Section 2, represented by its underlying graph, that is an out-tree. A tandem network is a special subclass of overtaking-free network, and it has been studied in [12] for the case of two queues and [13] for a general finite number of queues. Here, we extend those analyses to the whole class of overtaking-free networks.

The methodology consists in exploiting the product form expression of the stationary distribution of the network and comparing it with a closed one having the same number of customers inside as the one communicated to the tagged customer at the arrival epoch. We are allowed to make this comparison because all customers behind the tagged one can be overlooked under the overtaking-free assumption since they do not interfere in the calculation of their expected sojourn time.

Using this methodology, we compute a common equilibrium threshold strategy that all customers are going to implement in deciding whether to join or balk the system. This decision is only based on the partial information received at the arrival epoch, consisting of the total number of customers already inside the network.

The optimal strategy is computed by building an expected profit function P, that only depends on the number of users in the system and the common strategy followed by the rest of customers. The sign of the profit function corresponds to the decision of the tagged customer, they decide to join if the sign is positive, otherwise they decide to balk. Once they have joined the system, they are committed to their decision and cannot renege, and, if they choose not to join, they will not have another opportunity to come back.

Finally, we verify that this optimal response leads to an equilibrium.

Consider an overtaking-free network and denote with $G=(V,E)$ the associated out-tree graph. Let $V=\{1,2,\dots ,N\}$ and $E=\{e_2,e_3,\dots ,e_N\}$, so that N is the total number of queues, and $e_i$, for $i=2,\dots ,N$ denotes the edge entering in node i, which we know to be unique.

Customers arrive at the network through node 1, according to a Poisson process with intensity $\lambda$, and are served according to an FIFO discipline. A customer completing their service at node $i\in V$, is routed to another queue at node $j\in D\left(i\right)$ with probability $p_{ij}>0$. Remember that 0, the outside node, may belong to $D\left(i\right)$, allowing the customer to leave the network when served at node $i\in V$. The service times of customers in each queue $i\in V$ are independent and exponentially distributed with means $1/{\mu }_i$.

Figure 1 illustrates an example of an overtaking-free network.

4. Stationary Distribution

The only information that is provided to the customers at the moment of their arrival is the total number of customers in the system. Using this information, they may decide to join the network or balk, and they may make this decision according to a potentially random strategy. It follows that we may represent a strategy, say s, as an infinite vector, that is, $s=(s_0,s_1,\dots )$, where $s_k$ is the probability that an arriving customer will join the system after observing a total number of k customers in it. In particular, for $K\in \mathbb{N}$, we denote by ${\sigma }_K$ the K-threshold strategy, with the property that $s_k={\mathrm{𝟙}}_{\{k<K\}}$.

Remark 1. 

We say that a strategy s is a pure strategy whenever $s_k\in \{0,1\}$ for all $k\in \mathbb{N}$. A K-threshold strategy is a pure strategy. We say that a strategy s is a mixed x-threshold strategy, where $x=n+p$ with $n\in \mathbb{N}$ and $p\in (0,1)$, if $s_k=1$ for $k=0,\dots ,n-1$, $s_n=p$, and $s_k=0$ for $k>n$. This means that an arriving customer certainly joins if there are less than n customers in the system, joins with probability p if there are exactly n customers, and otherwise they balk. A mixed threshold strategy is a convex combination of two pure threshold strategies, indeed ${\sigma }_x=(1-p){\sigma }_n+p{\sigma }_{n+1}$.

We say that a strategy is admissible if whenever $s_K=0$ for a given $K\in \mathbb{N}$, then $s_k=0$ for any $k>K$. In this case, the number of customers in the network can be at most K, and the network is called semi-open. If the strategy s is positive, that is, $s_k>0$ for all $k\in {\mathbb{N}}_0$, the network is said to be open, and we set $K=\infty$. To simplify the exposition, in the following when we say strategy, we implicitly mean admissible strategy, and we denote by $\left|s\right|=K\in \mathbb{N}\cup \{\infty \}$ the maximum number of customers allowed in the network given the strategy s.

Under the assumption that everyone uses the common strategy s, we denote by $P\left(k\right|s)$ the expected potential profit function for an arriving customer that observes k users in the network, that is,

$$P\left(k\right|s)=R-C\left(k\right|s)=R-\sum _{i\in V}{C}_i{T}_i\left(k\right|s),$$

(1)

where R is the positive reward for entering the system and $C\left(k\right|s)={\sum }_{i\in V}{C}_i{T}_i\left(k\right|s)$ is the expected cost for the sojourn time. $C_i$ is the positive cost per unit sojourn time in queue i, also called the cost rate, and $T_i\left(k\right|s)={\mathbb{E}}_s\left[W_i\right|Q_a=k]$ is the expected sojourn time that a joining customer will spend in queue i under the common strategy s and having received the information $Q_a=k$. $W_i$ is the corresponding sojourn time and $Q_a$ denotes the total number of customers at their arrival time. The profit for balking is assumed to be 0.

In Section 5, we give an algorithm to compute $P\left(k\right|s)$, and find the equilibrium strategy by maximizing it under the assumption that all customers use the same strategy.

Thanks to the assumptions of independence and exponential service times, we can describe the dynamics of the network only by keeping track of the number of customers at each queue, that is, the vector $\overrightarrow{Q}\left(t\right)=(Q_i\left(t\right),i\in V)$. Indeed, the system can be described as a continuous-time Markov chain with values in ${\mathbb{N}}_0^{\left|V\right|}$. In particular, this kind of network can be seen as a generalized Jackson network, whose special property consists in having a state-dependent arrival rate due to the joining strategy s; according to the nomenclature in [23], this network belongs to the class of Whittle networks. For a background on Jackson networks, consult chapter 2 of [24], and for more general Whittle networks, the book [23]. In our case, the arrival rate is equal to $\lambda \left(t\right)=s_{Q\left(t\right)}\lambda$, where $Q\left(t\right)=|\overrightarrow{Q}\left(t\right)|$ is the total number of customers in the system, and $\lambda$ is the maximum arrival rate, the one achieved if no customers decide to balk.

A nice property of Jackson and Whittle networks is that the stationary distribution, when it exists, has the so-called product form, that is, the queues behave like they were almost independent.

We denote by $\overrightarrow{\nu }=({\nu }_i,i\in V_0)$ the solution of the normalized traffic equations, satisfying the normalizing condition ${\nu }_0=1$ and the following relation:

$${\nu }_i=\sum _{j\in V_0}{\nu }_j{p}_{ji},i\in V,$$

(2)

and, to simplify notations, we set $\overrightarrow{\eta }=({\eta }_i,i\in V_0)$, with ${\eta }_i={\nu }_i/{\mu }_i$.

When the network is overtaking-free, and its graph is an out-tree, the solution of the traffic Equation (2) is readily written. Indeed, denoting by $E_i\subset E$ the set of edges that belong to the unique path from node 1 to node $i\in V$, then ${\nu }_1=1$ and, for $i\ne 1$, ${\nu }_i={\prod }_{e\in E_i}p\left(e\right)$, where $p\left(e\right)=p_{ij}$ for $e=(i,j)$.

Let ${\overrightarrow{Q}}^{*}=(Q_i^{*},i\in V)$ be a vector distributed as the stationary number of customers in the network, and let ${\overrightarrow{\pi }}_s=\left({\pi }_s\left(\overrightarrow{n}\right),\overrightarrow{n}\in {\mathbb{N}}_0^{\left|V\right|}\right)$ be the stationary distribution, so that ${\pi }_s\left(\overrightarrow{n}\right)=\mathbb{P}({\overrightarrow{Q}}^{*}=\overrightarrow{n})$, with $\overrightarrow{n}=(n_i,i\in V)\in {\mathbb{N}}_0^{\left|V\right|}$ denoting a general state.

It is known that in a Whittle network, with any topological structure, at most one transition may occur at a time, and that the time between two transitions is exponentially distributed with a state-dependent rate. We define by ${\mathcal{T}}_{ij}$ the operator that moves one customer from node $i\in V_0$ to node $j\in V_0$, that is, ${\mathcal{T}}_{ij}\overrightarrow{n}=\overrightarrow{n}-{\overrightarrow{\mathbb{e}}}_i+{\overrightarrow{\mathbb{e}}}_j$, where ${\overrightarrow{\mathbb{e}}}_i$, for $i\in V$, is the $\left|V\right|$-dimensional unit vector with the i-th coordinate equal to 1, whereas ${\overrightarrow{\mathbb{e}}}_0=\overrightarrow{0}$. The transition rate from $\overrightarrow{n}\in {\mathbb{N}}_0^{\left|V\right|}$ to ${\mathcal{T}}_{ij}\overrightarrow{n}\in {\mathbb{N}}_0^{\left|V\right|}$, whenever $j\in D\left(i\right)$, is

$$q_s(\overrightarrow{n},{\mathcal{T}}_{ij}\overrightarrow{n})=\left\{\begin{array}{cc}{s}_{|\overrightarrow{n}|}\lambda p_{0j},\hfill & i=0\hfill \\ {\mu }_i{p}_{ij},\hfill & i\in V\hfill \end{array}\right.$$

In an overtaking-free network, $p_{01}=1$; however, we keep the notation general as the stationary regime holds for a more general topology.

Assuming that the stationary distribution exists, as occurs when the network is stable, it has the following form:

$${\pi }_s\left(\overrightarrow{n}\right)\propto S\left(\right|\overrightarrow{n}\left|\right){\lambda }^{|\overrightarrow{n}|}\prod _{i\in V}{\eta }_i^{n_i},|\overrightarrow{n}|\le |s|.$$

(3)

In the following, we will not check the stability assumption as it always holds when $\left|s\right|<\infty$. In (3), $S\left(n\right)={\prod }_{k=0}^{n-1}{s}_k$, and the missing normalizing constant is

$${\kappa }_s^{-1}=\sum _{|\overrightarrow{n}|\le |s|}S\left(\right|\overrightarrow{n}\left|\right){\lambda }^{|\overrightarrow{n}|}\prod _{i\in V}{\eta }^{n_i}.$$

To check that (3) is verified, one can check that the global balance conditions are satisfied; see also example 1.49 in [23].

In the following, we also deal with networks whose number of customers is kept constant, say $K\in \mathbb{N}$. In this case, the network is called closed, and in it, transitions from/to node $0\in V_0$ are not allowed; therefore, the strategy employed by all customers is irrelevant. When the network is closed, the normalizing condition in (2) is substituted with ${\nu }_1=1$. A closed network is always stable, and its stationary distribution has the following product form (compare with (3)):

$${\pi }^{\mid K}\left(\overrightarrow{n}\right)\propto \prod _{i\in V}{\eta }_i^{n_i},\left|\overrightarrow{n}\right|=K.$$

(4)

Denoting by ${\overrightarrow{Q}}^{\mid K}$ a random vector distributed according to this stationary distribution, it has the same joint distribution as $\left|V\right|$ independent geometric random variables with corresponding parameters ${\left({\eta }_i\right)}_{i\in V}$ conditioned to have a total sum equal to K.

Jackson and Whittle networks enjoy the MUSTA property (moving unit sees a time average) at simple network transitions (see example 4.38 in [23]), according to which, any customer moving among the queues or from/to the outside sees, before joining the destination node, a stationary network having one less than the maximum number of customers.

To better formalize this property, we need to introduce the Palm probabilities. We consider the stationary network process $({\overrightarrow{Q}}^{*}\left(t\right),t\in \mathbb{R})$, whose trajectories belong to the space $\mathbb{D}$ of right-continuous functions with left limits with values in ${\mathbb{N}}_0^{\left|V\right|}$, and for fixed $i,j\in V_0$ with $i\ne j$, we define the set $\mathcal{T}$ of simple network transitions from i to j as

$$\mathcal{T}=\{({\overrightarrow{Q}}^{*}\left(t\right),t\in \mathbb{R})\in \mathbb{D}:{\overrightarrow{Q}}^{*}\left(0\right)={\mathcal{T}}_{ij}{\overrightarrow{Q}}^{*}(0-)\}.$$

(5)

Assuming the common strategy s, we denote by ${\overrightarrow{Q}}^{\mid \mathcal{T}}=({\overrightarrow{Q}}^{*}\left(0\right)-{\overrightarrow{\mathbb{e}}}_j\mid \mathcal{T})$ the state of the network at time 0 just before a customer, making a $\mathcal{T}$-transition, has joined their destination node, and we define its distribution as ${\overrightarrow{\pi }}_s^{\mid \mathcal{T}}$. This distribution takes the name of a Palm distribution; it has the special property to be conditioned to the occurrence of a $\mathcal{T}$-transition that is a null event, and it was first introduced by Palm in [25].

We give a heuristic proof of the MUSTA property in the following proposition; for a more formal proof see [23] (Theorem 4.37 and Example 4.38).

Proposition 1 (MUSTA property).

Fix $i,j\in V_0$, $i\ne j$, and consider $\mathcal{T}$ as in (5).

For a (semi-)open network employing the strategy s, and with $s^{\prime }$ denoting the shifted strategy, that is, $s_k^{\prime }=s_{k+{\mathrm{𝟙}}_{\{i\ne 0\}}}$, then it follows that

$${\overrightarrow{\pi }}_s^{\mid \mathcal{T}}={\overrightarrow{\pi }}_{s^{\prime }},$$

(6)

where ${\overrightarrow{\pi }}_{s^{\prime }}$ denotes the distribution of a similar network in which all customers employ the strategy $s^{\prime }$.

If the stationary distribution ${\overrightarrow{\pi }}_s$ refers to a closed network with a number of customers equal to K, in which the strategy s is irrelevant, the distribution ${\overrightarrow{\pi }}_{s^{\prime }}$ refers to a closed network with $K-1$ customers.

Proof. 

To simplify notation, we set ${\eta }_0=1$ and $n+{\mathrm{𝟙}}_{\{i\ne 0\}}=|\overrightarrow{n}+{\overrightarrow{\mathbb{e}}}_i|$, for $\overrightarrow{n},\overrightarrow{n}+{\overrightarrow{\mathbb{e}}}_i\in {\mathbb{N}}^{\left|V\right|}$ and $|\overrightarrow{n}+{\overrightarrow{\mathbb{e}}}_i|\le |s|$. To compute ${\pi }_s^{\mid \mathcal{T}}\left(\overrightarrow{n}\right)$, we perform the following calculations:

$$\begin{array}{cc}\hfill {\pi }_s^{\mid \mathcal{T}}\left(\overrightarrow{n}\right)& =\underset{t\downarrow 0}{\lim }{\displaystyle \frac{\mathbb{P}({\overrightarrow{Q}}^{*}\left(t\right)=\overrightarrow{n}+{\overrightarrow{\mathbb{e}}}_j,{\overrightarrow{Q}}^{*}(0-)=\overrightarrow{n}+{\overrightarrow{\mathbb{e}}}_i)}{{\displaystyle \sum _{|\overrightarrow{m}+{\overrightarrow{\mathbb{e}}}_i|\le |s|}\mathbb{P}({\overrightarrow{Q}}^{*}\left(t\right)=\overrightarrow{m}+{\overrightarrow{\mathbb{e}}}_j,{\overrightarrow{Q}}^{*}(0-)=\overrightarrow{m}+{\overrightarrow{\mathbb{e}}}_i)}}}\hfill \\ \hfill & =\underset{t\downarrow 0}{\lim }{\displaystyle \frac{\mathbb{P}({\overrightarrow{Q}}^{*}(0-)=\overrightarrow{n}+{\overrightarrow{\mathbb{e}}}_i)(q_{ij}t+o\left(t\right))}{{\displaystyle \sum _{|\overrightarrow{m}+{\overrightarrow{\mathbb{e}}}_i|\le |s|}\mathbb{P}({\overrightarrow{Q}}^{*}(0-)=\overrightarrow{m}+{\overrightarrow{\mathbb{e}}}_i)(q_{ij}t+o\left(t\right))}}}\hfill \\ \hfill & ={\displaystyle \frac{{\overrightarrow{\pi }}_s(\overrightarrow{n}+{\overrightarrow{\mathbb{e}}}_i)}{{\displaystyle \sum _{|\overrightarrow{m}+{\overrightarrow{\mathbb{e}}}_i|\le |s|}{\overrightarrow{\pi }}_s(\overrightarrow{m}+{\overrightarrow{\mathbb{e}}}_i)}}}={\displaystyle \frac{S(n+{\mathrm{𝟙}}_{\{i\ne 0\}}){\lambda }^{n+{\mathrm{𝟙}}_{\{i\ne 0\}}}{\prod }_{k\in V}{\eta }_k^{n_k}{\eta }_i}{{\displaystyle \sum _{|\overrightarrow{m}+{\overrightarrow{\mathbb{e}}}_i|\le |s|}S(m+{\mathrm{𝟙}}_{\{i\ne 0\}}){\lambda }^{m+{\mathrm{𝟙}}_{\{i\ne 0\}}}\prod _{k\in V}{\eta }_k^{m_k}{\eta }_i}}}\hfill \\ \hfill & ={\displaystyle \frac{S^{\prime }\left(n\right){\lambda }^n{\prod }_{k\in V}{\eta }_k^{n_k}}{{\displaystyle \sum _{|\overrightarrow{m}|\le |s^{\prime }|}{S}^{\prime }\left(m\right){\lambda }^m\prod _{k\in V}{\eta }_k^{m_k}}}}={\overrightarrow{\pi }}_{s^{\prime }}\left(\overrightarrow{n}\right)\hfill \end{array}$$

where $S^{\prime }\left(n\right)={\prod }_{k=0}^{n-1}{s}_k^{\prime }$. To obtain the result for a closed network with K customers, it is enough to repeat the same reasoning above, after setting $S\left(n\right)={\lambda }^{-n}$, considering $|\overrightarrow{n}+{\overrightarrow{\mathbb{e}}}_i|=K$ and summing over $\left|{\mathcal{T}}_{ij}^{-1}\overrightarrow{m}\right|=K$. □

When $i=0$, the $\mathcal{T}$-transitions correspond to the Poisson arrivals. This is the most recurrent case, and the property takes the well-known name of PASTA (Poisson arrivals see time averages); see [26]. We collect this fundamental result in the following corollary.

Corollary 1 (PASTA property).

In a stationary regime, an arrival customer sees an open network in a state distributed according to the stationary distribution, that is,

$${\overrightarrow{Q}}_a\sim {\overrightarrow{Q}}^{*}.$$

(7)

If the network is closed (semi-open) with K (maximum) customers, $Q^{*}$ in (7) refers to the stationary distribution of a similar network with $K-1$ (maximum) customers.

Thanks to the PASTA property, $T_i\left(k\right|s)={\mathbb{E}}_s\left[W_i\right|Q^{*}=k]$, where $Q^{*}=\left|{\overrightarrow{Q}}^{*}\right|$. In applying this formula, it is important to interpret the random variable $W_i$ as the sojourn time that a new customer, joining the queue $j\in V$ with probability $s_k{p}_{0j}$ (in an overtaking-free network $j=1$), is going to spend in queue $i\in V$ along their path through the network. The total number of customers in the network, in case and after the new one has joined, is then $k+1$.

Another property of a Jackson network with a state-dependent arrival rate consists in the fact that under the partial information $Q^{*}=k$ its stationary distribution coincides with that of a similar closed network with k circulating customers. We collect this well-known result in the following proposition, noticing that its proof is an immediate consequence of the product form (3) and the definition of the conditional probability.

Proposition 2. 

With ${\overrightarrow{Q}}^{\mid k}$ being a random vector distributed as the stationary number of customers in a closed network with $k\in {\mathbb{N}}_0$ total customers, the following equality in distribution holds

$$\left({\overrightarrow{Q}}^{*}|Q^{*}=k\right)\sim {\overrightarrow{Q}}^{\mid k}.$$

(8)

The next proposition gives a known efficient algorithm to compute the expected number of customers, $L_i\left(k\right)=\mathbb{E}\left[Q_i^{\mid k}\right]$, at each node $i\in V$; see Section 2.5.5 in [24] for a proof.

Proposition 3 (Mean value analysis).

The mean number of customers at node $i\in V$ in a closed queueing network with $K\in \mathbb{N}$ customers can be computed recursively according to the following algorithm:

$$L_i\left(K\right)={\displaystyle \frac{K{\eta }_i(L_i(K-1)+1)}{{\sum }_{j\in V}{\eta }_j(L_j(K-1)+1)}},$$

(9)

with $L_i\left(0\right)=0$.

Corollary 1 and Proposition 2 imply that an arriving customer that sees k customers in the system behaves in the same way as an arriving customer belonging to a closed network with $k+1$ customers. That is, before the arriving customer joins, the customers in the network are distributed as ${\overrightarrow{Q}}^{\mid k}$, and just after joining they are distributed as ${\overrightarrow{Q}}^{\mid k+1}$.

Since the tagged customer is indistinguishable from all others, we can use the information about the distribution of the customers in the system to finally estimate their expected sojourn time in each queue.

Theorem 2. 

In an overtaking-free network, assuming that the tagged customer only knows the total number of customers in the network at their arrival epoch, say k, independently of the common strategy s, their expected sojourn time at node $i\in V$ is equal to

$$T_i\left(k\right)={\eta }_i(1+L_i\left(k\right)),$$

(10)

and it can be computed recursively by the formula $T_i\left(0\right)={\eta }_i$ and, for $k\in \mathbb{N}$,

$$T_i\left(k\right)={\eta }_i\left(1+{\displaystyle \frac{k{T}_i(k-1)}{{\sum }_{j\in V}{T}_j(k-1)}}\right).$$

(11)

Proof. 

At the arrival epoch of the tagged customer, by the PASTA property (7) they encounter a stationary number of users in the network. Once they receive the information that the total number of users in the network is equal to k, the distribution of the customers in the network reduces to that of a stationary closed network with k customers, according to Proposition 2.

After that moment, and assuming that they decide to join, by the overtaking-free condition the future arrivals will not interfere with their future sojourn times they will spend in each queue $i\in V$ along their random path through the network. Therefore, we are allowed to change the future arrivals, so that we may assume that the network evolves as a closed network with $k+1$ customers in it. The tagged customer is then indistinguishable with respect to any other customer of this new closed network.

Looking now at the tagged customer as a typical customer in a closed network with $k+1$ customers, an application of the MUSTA property (6) implies that at any transition they will meet a network whose customers are distributed as a closed network with k customers inside. This allows the computation of their expected sojourn time. Indeed, in a stationary closed network with k customers, these customers are distributed according to the distribution given in (4).

The tagged customer visits the node $i\in V$ with probability ${\nu }_i$, and they spend there a sojourn time equal to $(1+Q_i)/{\mu }_i$. We obtain that $W_i=B{\sum }_{k=1}^{1+Q_i}{E}_k$, with B being an independent Bernoulli random variable with parameter ${\nu }_i$ and ${\left\{E_k\right\}}_{k>0}$ being independent exponentially distributed random variables with parameter ${\mu }_i$. By taking the expectation, (10) readily follows. Then, using this formula and (9), we obtain (11). □

Remark 2. 

A short observation about the use of the overtaking-free property in the proof of Theorem 2 is due, as it is subtle. Since the MUSTA property (6) holds for any Jackson and Whittle network, one would be tempted to believe that the overtaking-free condition is not required, and that in a general network in which overtaking is allowed, one could easily compute the distribution or the expectation of the waiting times at each queue. The observation that should be made is that, in this case, the tagged customer would no longer be an indistinguishable customer belonging to their network. Indeed, at the moment of their entrance, they had more information with respect to the others, that is, they knew that in the network there were exactly k customers circulating. However, they do not even belong to a $k+1$ closed network because, due to the permitted overtaking, the number of customers ahead of them may change due to some later arrival. Therefore, we are not allowed to apply the MUSTA property in the way we have in the proof of the theorem.

Remark 3. 

The result in Theorem 2 extends the one in [13], limited to tandem queues, to the more general class of overtaking-free Jackson networks with rate-dependent arrivals. In [13], the strategy to compute the expected sojourn time in each queue consists in using Little’s law, that allows us to obtain the value of ${\mathbb{E}}_{{\sigma }_K}\left[W_i\right]$ for any $K\ge 0$. Then, by using the relation

$$T_i\left(K\right|s_{K+1})={\displaystyle \frac{{\mathbb{E}}_{{\sigma }_{K+1}}\left[W_i\right]}{{\mathbb{P}}_{{\sigma }_{K+1}}(Q^{*}=K)}}-{\displaystyle \frac{{\mathbb{P}}_{{\sigma }_{K+1}}(Q^{*}<K)}{{\mathbb{P}}_{{\sigma }_{K+1}}(Q^{*}=K)}}{\mathbb{E}}_{{\sigma }_{K+1}}\left[W_i\right|Q^{*}<K],$$

(12)

(Equation (4) in [13]), one can finally obtain the result given in (10). However, we point out that Little’s law is very robust, meaning that it works in almost any queueing context, practically regardless of any distributional condition. This may misleadingly lead us to believe that a similar result holds for more general tandem networks, or even for overtaking-free networks. However, this is not the case; see Remark 4. Indeed, to make the expression in (12) calculable, it is still required that ${\mathbb{E}}_{{\sigma }_{K+1}}\left[W_i\right|Q^{*}<K]={\mathbb{E}}_{{\sigma }_K}\left[W_i\right]$. This equality only holds for very specific models, such as Jackson and Whittle networks, that are very restrictive in terms of the distributions of their structural random variables. As a byproduct of the special properties of these networks, one also obtains that ${\mathbb{P}}_{{\sigma }_{K+1}}(Q^{*}=K)/{\mathbb{P}}_{{\sigma }_{K+1}}(Q^{*}<K)={\mathbb{P}}_{{\sigma }_K}(Q^{*}=K)$.

Remark 4. 

To be convinced that Theorem 2 makes strong use of the assumption of exponentially distributed service times, we recall here some counterexamples. In particular, a single queue may always be seen as a tandem network with a single node, and consequently, also as an overtaking-free network. In the literature, the single queue has been extensively studied, and in particular, ref. [27] has shown an example of an M/G/1 queue that does not admit optimal equilibrium strategies. Moreover, the $M/G/1$ queue has been analyzed in full generality in [28], where it is shown that, depending on the service distribution, the equilibrium is not necessarily unique, and either the avoid-the-crowd phenomenon or the follow-the-crowd phenomenon may occur.

5. Equilibrium Strategies

From an economic perspective, customers will join or balk the network according to an evaluation of their expected profit. Therefore, to look for the existence of an optimum equilibrium strategy, we need to study the expected profit function, $P\left(k\right)$ in (1), of a tagged customer observing, at the arrival epoch, that the network already hosts k customers. We have omitted in the notation the explicit reference to the common strategy s, as it is actually independent of it, as shown more clearly below. The customer decides to join the system if $P\left(k\right)>0$ and balks if $P\left(k\right)<0$, while they may indifferently take either action when $P\left(k\right)=0$. This suggests that an optimal strategy would be a threshold one, that is, ${\sigma }_K$, where K is the turning point of the sign of the function $P\left(k\right)$. In this section, we show that, indeed, this threshold strategy leads to an equilibrium.

We start by studying the monotonicity properties of the cost functions $T_i\left(k\right|s)$ appearing in (1) and computed in Theorem 2 by exploiting the ordering relations satisfied by the random variables $Q_i^{\mid k}$ as functions of k.

Lemma 1 (Stochastic ordering).

The vectors ${\left\{{\overrightarrow{Q}}^{\mid k}\right\}}_{k\in {\mathbb{N}}_0}$ are stochastic ordered in $k\in {\mathbb{N}}_0$. That is, for any function $f:{\mathbb{R}}^{\left|V\right|}\to \mathbb{R}$, non-decreasing in each coordinate,

$$\mathbb{E}[f({\overrightarrow{Q}}^{\mid k+1})]\ge \mathbb{E}[f({\overrightarrow{Q}}^{\mid k})].$$

Proof. 

The proof relies on a coupling analysis, that is, we construct on the same probability space two closed networks with $k\in {\mathbb{N}}_0$ and $k+1$ customers, and show that the number of customers in each queue in the latter network is at least equal to the corresponding one in the former. The coupling is achieved by considering a closed network with $k+1$ customers inside, k of white color and one of red color. We also assume that the order of service is FIFO for the white customers, and preemptive resuming with low priority for the red customer. That is, the red customer is always pushed back to the end of the queue when there are some white customers at the same node, and also when the red customer is the one being served, being alone, and a white customer joins the same node. It follows that the dynamics of the white customers are not affected by the presence of the red customer; in other words, if we count only them in the network, they behave exactly as a closed network with k customers inside. On the other side, if we consider all customers, regardless of their color, since the service times are exponentially distributed, they behave exactly as a closed queueing network of $k+1$ customers. Since the number of customers in each queue of the latter network is at least equal to the one in the former, counting only the white customers, the result follows. □

The monotonicity property proved in Lemma 1 immediately implies the following result on the expected sojourn times. It extends and corrects the results given in [12]; see Appendix B.

Theorem 3 (Monotonicity of the expected sojourn time).

For $i\in V$, the expected sojourn time in queue i, $T_i\left(k\right|s)$ is strictly increasing and $T_i\left(k\right|s)\to +\infty$ as $k\to \infty$.

Proof. 

The result follows by showing that for $i\in V$ the variables ${\left\{\left(W_i\right|Q_a=k)\right\}}_{k\ge 0}$ are strictly stochastically ordered in k. We have that $\left(W_i\right|Q_a=k)\sim B{\sum }_{k=1}^{1+Q_i^{\mid k}}{E}_k$, with B an independent Bernoulli random variable, with parameter ${\nu }_i$ and ${\left\{E_k\right\}}_{k>0}$ independent exponentially distributed random variables with parameter ${\mu }_i$. Since $W_i$ is an increasing function of $Q_i^{\mid k}$, the result follows by the stochastic order of the random variables ${\left\{Q_i^{\mid k}\right\}}_{k\ge 0}$ proved in Lemma 1. The order is strict as $P(Q_i=k+1|Q_a=k+1)>0$, while $P(Q_i=k+1|Q_a=k)=0$ for any $k\ge 0$. The last result mentioned in the statement follows by applying L’Hopital’s rule to the limit of expression (10) as $k\to \infty$. □

By Theorem 2 and (1), we can compute the expected profit function $P\left(k\right)=P\left(k\right|s)$ in the following way:

$$P\left(k\right)=R-\sum _{i\in V}{C}_i{T}_i\left(k\right)k\in {\mathbb{N}}_0.$$

(13)

As anticipated, the profit function $P\left(k\right)$ does not depend on the strategy s and of the arrival rate $\lambda$. Furthermore, by Theorem 3, $P\left(k\right)$ is strictly decreasing in k and ${\lim }_{k\to +\infty }P\left(k\right)=-\infty$, which means that the benefit of joining the network decreases as a function of the customers that are already inside.

It follows that at some point the function $P\left(k\right)$ becomes negative, and we use this fact to show that there exists a common strategy, in particular in the class of threshold ones, that is optimal for all customers to implement.

Theorem 4 (Equilibrium strategies).

Let $K=\min \{k\in {\mathbb{N}}_0:P\left(k\right)<0\}$ be the threshold at which the profit function becomes negative. If $P(K-1)>0$, ${\sigma }_K$ is the only pure equilibrium threshold strategy. If $P(K-1)=0$, the only equilibrium strategies are all the mixed threshold strategies ${\sigma }_{K-1+p}$, with $p\in [0,1]$.

Proof. 

Assume that everyone else is using the strategy s, and consider a tagged customer that follows the strategy $\tilde{s}$. Then, their payoff, as a function of the customers they find in the network, is given by

$$\tilde{P}\left(k\right)={\tilde{s}}_k\times P\left(k\right|s)={\tilde{s}}_k\times P\left(k\right),$$

and does not depend on the common strategy s. Therefore, by optimizing their profit, we obtain that the optimal strategy ${\tilde{s}}^{*}$ satisfies ${\tilde{s}}_k^{*}={\mathrm{𝟙}}_{\{k<K\}}$ for $k\ne K-1$ and, assuming $K>0$,

$$\begin{array}{ccc}{\tilde{s}}_{K-1}^{*}\hfill & =1\hfill & \mathrm{if} P(K-1)>0,\hfill \\ {\tilde{s}}_{K-1}^{*}\hfill & \in [0,1]\hfill & \mathrm{if} P(K-1)=0.\hfill \end{array}$$

(14)

That is, $\tilde{s}={\sigma }_K$ is an optimal strategy, and since it is optimal independently of the common strategy s, it is also optimal against itself. In addition, if $K>0$ and $P(K-1)=0$, then also the strategy ${\sigma }_{K-1}$ is optimal and an equilibrium strategy, as well as any convex combination of ${\sigma }_{K-1}$ and ${\sigma }_K$. □

6. Numerical Computations

The easiest example of an overtaking-free network that is not a tandem network has an underlying out-tree $G(V,E)$ with $V=\{1,2,3\}$ and $E=\left\{\right(1,2),(1,3\left)\right\}$. We compare this network with a two-node tandem network by looking at the equilibrium threshold strategies for different values of waiting costs per unit of time, as the routing probabilities vary. We refer to the first network as model A and to the two-node tandem network as model B.

As Figure 2 shows, we choose for model A the routing probabilities $p_{12}=p_{13}=1/2$, which imply ${\nu }_1=1$ and ${\nu }_2={\nu }_3=1/2$, and all service rates are equal to $\mu$. For model B (see [12]), we choose ${\mu }_1=\mu$ and ${\mu }_2=2\mu$, ${\nu }_1={\nu }_2=1$.

The first node of the two models is equal; however, in model A, after completing service in queue 1, customers are routed to queue 2 or queue 3 with equal probability and served there at rate $\mu$, while the customers in model B are all served in queue 2 but with twice the service rate, that is, $2\mu$. In both systems, customers pay a cost C per unit of sojourn time in any queue.

We expect the equilibrium strategies to be similar to each other, due to the fact that even if in model A queues 2 and 3 work at a speed that is half that of queue 2 in model B, they should handle, in the long run, half of the customers each.

Since both models have a simple topology, we can obtain explicit formulas for the profit function as expressed by the following theorem.

Theorem 5. 

The profit functions for models A and B are, respectively, equal to

$$\begin{array}{cc}\hfill P_A\left(k\right)& =R-{\displaystyle \frac{(k+1)C}{\mu }}·{\displaystyle \frac{2^{k+3}-(k+4)}{2^{k+3}-2(k+3)}}\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill P_B\left(k\right)& =R-{\displaystyle \frac{(k+1)C}{\mu }}·{\displaystyle \frac{2^{k+2}-1}{2^{k+2}-2}}\hfill \end{array}$$

(16)

Proof. 

For model A, we have that the vector $(Q_1^{\mid k},Q_2^{\mid k}+Q_3^{\mid k})\sim {\overline{(G_1,E_2)}}^k$ is given by the truncation of a bivariate vector made of two independent random variables with distributions $\mathrm{Geo}\left({\eta }_1\right)$ and $\mathrm{dErlang}(2,{\eta }_2)$. Since ${\eta }_2/{\eta }_1=1/2$, by Lemma A2, we have $Q_2^{\mid k}+Q_3^{\mid k}\sim {\overline{\mathrm{dErlang}(2,1/2)}}^k$, and by (A5),

$$m_E\left(k\right)=:\mathbb{E}[Q_2^{\mid k}+Q_3^{\mid k}]=2{\displaystyle \frac{(k+1)(k+4)-2^{k+3}+4}{2(k+3)-2^{k+3}}},$$

and, having that $T_A\left(k\right)=(k+2-m_E\left(k\right)/2)/\mu$, the relation (15) holds.

For model B, we have that the vector $(Q_1^{\mid k},Q_2^{\mid k})\sim {\overline{(G_1,G_2)}}^k$ is given by the truncation of a bivariate vector made of two independent Geometric random variables with parameters ${\eta }_1$ and ${\eta }_2$. By Lemma A1, $Q_1^{\mid k}\sim {\overline{\mathrm{Geo}\left(2\right)}}^k$, since ${\eta }_1/{\eta }_2=2$, and by (A1),

$$m_G\left(k\right)=:\mathbb{E}\left[Q_1^{\mid k}\right]={\displaystyle \frac{k+1}{p^{k+1}-1}}+k+{\displaystyle \frac{1}{1-p}},$$

and, having that $T_B\left(k\right)=(k+3-m_G\left(k\right))/\left(2\mu \right)$, the relation (15) holds. □

The functions $P_A\left(k\right)$ and $P_B\left(k\right)$ are graphically represented in Figure 3a, where we fixed $R=9$, $C=2$, and $\mu =1.15$. Figure 3b shows the equilibrium thresholds as functions of $\mu$. We observe that $P_A\left(k\right)$ is slightly lower than $P_B\left(k\right)$, and therefore, the equilibrium thresholds in model A are generally lower than the ones in model B. This means that the customers in model A are more intimidated by the presence of customers in the network and decide to stop to enter for a less crowded network than the ones in model B. For example, looking at $\mu =1.15$ in Figure 3a, we have that $P_A\left(4\right)<0<P_B\left(4\right)$, implying that a model A network would be filled by at most three customers while a model B network would allow up to four customers, as visible in Figure 3b for $\mu =1.15$. This is due to a well-known fact in queueing theory, indeed, model A is a less efficient network in that it wastes service capacity when one queue, among nodes 2 and 3, is idle while the other is working. Figure 4 shows a throughput comparison for the model A and model B networks as $\mu$ varies. As expected, the throughput of model B is higher than that of model A, and both, as $\mu \to \infty$, converge to their maximum value, which is equal to the arrival rate $\lambda =1$.

A quick comparison between model A and model B would suggest that model B is superior in achieving a higher throughput by keeping the same (limiting) average service rate of the second service. However, a deeper economic analysis would require more detailed considerations of the specific settings of the system at hand. In particular, if model B is already implemented with the second step having a service rate equal to $\mu$, it could be cheaper to add another system, so converting it to a model A network, rather than replacing the actual server with one with double the service rate.

In other practical situations, such as a healthcare center, the employment of a model B can reflect the fact that not all patients follow the same route, due, for example, to the fact that patients belong to different groups requiring different treatments.

Now, we take a closer look at a variant of model A by changing the routing probabilities and the costs per unit of sojourn time, in order to see how the equilibrium threshold is affected. We fix ${\mu }_i=i$ for $i=1,\dots ,3$, $R=20$, $C_1=1$, and we relax the assumption that $p_{12}=p_{13}$. We assume that $C_1+C_2+C_3=13$ (i.e., $C_3=12-C_2$), with $C_2\in [0,12]$, and that $p_{12}+p_{13}=1$ (i.e., $p_{13}=1-p_{12}$), with $p_{12}\in [0,1]$. The values of the equilibrium threshold are represented in Figure 5. We observe that the thresholds are higher for low $C_2$ cost values and high $p_{12}$ probability values as well as for high $C_2$ cost values and low $p_{12}$ probability values. On the other side, they are lower when both $C_2$ and $p_{12}$ values are either high or low, forming a sort of saddle point in the middle.

For example, reading from Figure 5a with $C_2=10$, or equivalently reading the brown line in Figure 5b, we have that increasing the value of $p_{12}$, the maximum number of people inside the network decreases. Indeed, increasing $p_{12}$ implies more use of node 2, whose service rate ${\mu }_2=2$ is lower than the service rate of node 3, ${\mu }_3=3$.

7. Limitations and Future Work

This research model has several limitations. First, the overtaking-free assumption, while simplifying the problem, may not reflect real-world scenarios. For example, in healthcare, patients often require multiple treatments, and service order may not align with arrival time. Relaxing this assumption, even within the context of Jackson networks, could lead to intriguing and challenging problems. A potential starting point would be analyzing small networks with limited nodes and few routing paths, similar to [15]. Second, the model assumes exponential service times, which may oversimplify real situations. As noted in Remark 4, even single-server systems with non-exponential service times can exhibit complex behavior, and this complexity would increase in queueing networks.

Future research could also explore models with other kinds of partial information. While fully observable and unobservable cases are relatively straightforward, real applications may need more refined forms of partial information, requiring deeper analysis. Lastly, incorporating heterogeneous user classes and priorities, even keeping the overtaking-free and exponential service assumptions, presents another promising direction for further research.

8. Conclusions

Overtaking-free networks are special networks with the property that no customers entering behind a tagged customer may influence their sojourn time in the network. Under the assumption of single-server queues and exponentially distributed inter-arrival and service times, these networks are characterized by having an out-tree structure of the underlying graph.

For these networks, under the assumption that a tagged customer is informed about how many customers are already in the network at their arrival epoch, we compute the equilibrium strategies, that are shown to be of threshold type. Indeed, this follows from the fact that under the partial information received by the arriving tagged customer, their sojourn times, as well as their profit function, do not depend on the common strategy used by the other customers.

Finally, we remark that the results heavily rely on the special property of the exponential distribution as there are known and easy to construct counterexamples by using the simple M/G/1 queue.